You mention balance, so possibly do one lesson CA and then one 'traditional methods' (the co-Ts way). After, give a survey to the students to see which they think is better (as they are the 'customer').
     You could 'mask' the survey in a lesson about different learning strategies. Students seem to see real value in recognising, and having a better understanding how learners can differ in their approaches to learning (a foreign language).
      Of course, if you don't like the results of the survey then possibly a mix methods approach should be considered.

Honestly, her approach sounds detrimental to student development and doesn't best promote their learning.  Implement the techniques and pedagogy you've learned to be beneficial as best you can reasonably do so while still maintaining a harmonious relationship.  Perhaps she might come around a bit if you focus on the key expressions and vocabulary from the textbook (along with other structures and vocab you think would be good to introduce) in the first unit of the lesson.  The textbook stuff is simple and kids can develop familiarity with it and beyond it in that unit, then she can hopefully be reassured that they have that knowledge basis the curriculum demands while you more readily move beyond it in the later units of the lesson.

Selfishly, the main concern is my own well-being.

Since the textbook was not designed with scaffolding in mind, meaning the authors gave no consideration to the most widespread, popular and effective technique in education for the past 100yrs, the only real way to teach the textbooks' vocabulary and Key Expressions is to do nothing but Classically approached drills centred around rote memorisation.

Doing that kind of shit day in and day out might not bother her, but it makes me utterly miserable.

It's not our opinion that her approach is outdated and wrong, it's the opinion of pretty much all educational research over the last 40-50yrs. Still, I'm not trying to change her approach, I just don't want her to interfere with mine.   

I have a co-t similar to this too, though she isn't worried about the parents, as she is retiring at the end of this semester.
So what we agreed upon is, I will teach the textbook stuff as is, unless it is really bizzare / or really wrong. Once we get to the bit where we are practicing the key expressions more, in my PPT I'll make the necessary changes for it to be used correctly. I'll usually touch on it when we ready over the study point at the beginning of the lesson, and remind them at the PPT and my co-t will explain it in Korean.
Then, outside of the textbook examples, I'll give them the correct way to use the key expressions that we are learning.

I typically use fill in the blanks, using enough pictures to get the point across, and they fill in the blanks from the pictures. I also like to use a bit more obscure things, for example, if I do a sport one, I might use jousting instead of soccer / tennis / etc. My students love this as they learn a new word, and often find the pictures for them funny. (Though, I likely wouldn't do this for elementary school).
I felt like this has striked a good balance with teaching what they need to know for their exams/tests, and teaching them how to use it correctly. If it's something that I can talk / use with them outside of class, I'll pretend not to understand the incorrect way then too, and that forces them to use it the way I taught them in class. I've gotten my students to stop saying "I'm fine/so-so" when I ask them how they are by doing this, and they sound a lot more natural when I ask them.

This makes me wonder what Korean teachers actually learn in their version of teachers' college. 

We have a group of trainee teachers in at the moment, so I discussed this with them yesterday. They cover the history of teaching methodology, and then move on to modern/current approaches; using the communicative approach as an umbrella term it would include things such as TBL, PBL, etc. Pretty much what any post-grad course will cover for TESOL.
The trainees said CA is the dominant approach within Korea, but as recognised within the literature and relevant studies (see Li,1998 for example), implementing a 'western' approach in Korea has issues, namely, L2 proficiency of the teacher.

Whilst admittedly dated now, Swan and Smith (2001) give a good (simplistic) outline on how English is/was taught/learnt in Korea, and shows why (traditionally) Korean English teachers hold preference for teacher-centred instruction.

Personally, I think the majority of Korean English Teachers are stuck between a rock and a hard place. They have the knowledge, ability, and willingness to implement a variety of approaches, but exams such as the Suneung treat English as a science, not as an art.
Therefore, the 'real-world uses' for English in Korea is to pass grammar-focused exams.

Before I decide what to do, I'd like to hear her thoughts on how she would like to be involved in the lesson. Ideally, I'd like to focus my lessons on using the Key Expressions, or a modified version of the Key Expressions if I decide to alter them to be less contrived, and her lessons can be focused around memorising the textbook's vocabulary.
 
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Apologies, I've only ever co-taught at summer/winter camps so I have relatively little experience managing relationships with co-ts. However, what you have mentioned above about understanding the level of collaboration between the two of you sounds like a good starting point.
It might be worthwhile for both of you to discuss your experiences,good and bad, of learning an L2. I found this to be a softer/indirect way of critiquing methods.

There's also zero discipline policy for public schools, no demerit system, no detention, nothing. Corporal punishment was made illegal in 2009, yes, but you can't outlaw the main discipline strategy and leave nothing in its place.

They do have a demerit system, those green mileage points. If you get too many of those, it doesn't look good on applications to high schools, or post-secondary institutions. Of course NETs are rarely given knowledge of this, and for those 'lost' students who don't care about finishing high school it doesn't help.

What kind of f*cked up educator would put that kind of thing on their student's permanent record? More importantly, what kind of idiot thought that system was a great idea?

Militaries and service academies function on such a system. Probably where it trickled down from. Not saying that was a good decision, just that's probably where it came from, which sort of makes sense at some point in the past of Korea's history and combine that with some person who got to where they were through relentless study and thinks that's how everyone thinks and voila- that's what you get. Maybe it made sense then in like the 60s. Now? Probably not so much.

Now, its effectiveness is dependent on how much the student body tends to value going to a good school. In Korea it might be more of a deterrent than you think. There are some real consequences to going to different high schools and real consequences at home and socially. A student who is facing not going to the high school all their friends are going to will have strong incentive to not earn any more demerits, also same if a girl they like is going there. There is also still some social status tied to academic achievement, so there is that.

At the same time of course, even if it is more effective, there are students with whom it will be wholly ineffective. It's sort of similar to detention- Good deterrent for good students, terrible deterrent and method for bad students. So yeah, the system isn't completely moronic, I'd just say it's not really effective by itself, and by not really effective, I'd say really not really effective (probably only effective for 10-25% of students, maybe 40% tops at high-level schools). It really needs other forms of discipline along with it if it is to work. Even militaries that use demerits also have other punishments such as revoking leave, menial duties, CP in the old days, etc.

In other words, the point is to help learners correct their behaviour; the learner sits in detention for an afternoon or is given extra homework, yes it sucks, but the reinforcement encourages them to control/correct their behaviour

What follows below is long-winded whathaveyou. You can skip to the final paragraph for the ultimate point.

As someone who frequently found himself in detention during one phase of my youth, I think for recidivists, it doesn't have much of a deterrent or corrective effect. For good students, certainly, but its not like anything super bad is happening so the deterrent effect isn't that great. I mean, in high school I once got In School Suspension for socking a kid in the face. One day of ISS. Did this do anything to make me regret what I did or deter me from getting in another fist fight? Not really. What changed was about a year later I finally realized I had been a bit of a dick that day. I'd say either personal counseling or brutal physical labor are better deterrents. Something that really makes you stop and think before you do something OR really tries to address the problem.

Deterring bad students with detention is like deterring crime with having to spend a night in a holding cell. That's not really a deterrent. Heck, holding (and detention) were both almost like a party sometimes. I mean, in some cases you just get to chill out and relax. Other times everyone is laughing and joking and having a blast. Also, the person in charge of detention, a lot like the COs, is probably talking and bsing with you too. I mean, it's not like teachers volunteer for detention. They get assigned to do it. You get some teacher having to stay an extra hour or two on a Friday afternoon while the rest are off to whatever TGIF they have going on, they're going to want the time to fly-by as well.

Also, if you have an optimistic personality, it's really hard to make something a punishment sometimes. Such a person will always find something good about the situation. Like, when I had lunch detention, I got extra food and it was kind of fun doing dishes in a weird way. Do good honest work and try and figure out how to do a task efficiently and so on. Get to learn stuff. Then there was the detention where I had to copy Bible pages. That was good- Same as jail: You do spend some time with the Good Book. Dictionary page copying detention? Alright, learn some new words. Just sit there quietly detention? Alright, either sleep and I have a good imagination I can entertain myself. Bored Teacher detention? Spend the hour talking. ISS? All the fun people are there from time to time- and you get to chill with Dr. Abdul-Aziz and he'll talk to you about life (probably actually the one good, corrective part of detention was talking about life with Dr. A-A). I mean, none of those really worked. I never was a super delinquent or did anything habitually bad, but yeah, regular busts for petty stuff, and you're in and out.

What is the solution then? I really think a mixed approach is what you need. Students are different. If some relentlessly driven academic type is threatened with demerits, that could have a real effect. On the other hand, for students who are really energetic, detention could be effective. For students who are physically lazy, some kind of work or exercise might be effective. And for students who seem to only respond to force, CP might be appropriate.

We have a group of trainee teachers in at the moment, so I discussed this with them yesterday. They cover the history of teaching methodology, and then move on to modern/current approaches; using the communicative approach as an umbrella term it would include things such as TBL, PBL, etc. Pretty much what any post-grad course will cover for TESOL.
The trainees said CA is the dominant approach within Korea, but as recognised within the literature and relevant studies (see Li,1998 for example), implementing a 'western' approach in Korea has issues, namely, L2 proficiency of the teacher.

Whilst admittedly dated now, Swan and Smith (2001) give a good (simplistic) outline on how English is/was taught/learnt in Korea, and shows why (traditionally) Korean English teachers hold preference for teacher-centred instruction.

Personally, I think the majority of Korean English Teachers are stuck between a rock and a hard place. They have the knowledge, ability, and willingness to implement a variety of approaches, but exams such as the Suneung treat English as a science, not as an art.
Therefore, the 'real-world uses' for English in Korea is to pass grammar-focused exams.

Well, fair enough. 

However, I never read or hear about ANY Korean teacher working and communicating with the NET and explaining this to them. As you said, they have the ability.  So, that just leads me to believe they are either lazy, incompetent, resentful or just love setting up NETS to fail, in their own passive-aggressive way(s). 

The trainees said CA is the dominant approach within Korea, but as recognised within the literature and relevant studies (see Li,1998 for example), implementing a 'western' approach in Korea has issues, namely, L2 proficiency of the teacher.

This is very true, though I still doubt CA is the "dominant approach within Korea". In all my years I've yet to see the English language divided into competencies and any of them taught in a CA manner. Perhaps it's the dominant approach that's taught
at university, but even then... it makes no sense.

No domestic educational degree wouldn't take into account the national curriculum. Korea's English curriculum, if you can call it that, couldn't be more incompatible with the CA. If universities are going to maintain that CA is the dominant pedagogical approach that is taught then it's all for show.

L2 proficiency is a big issue though. My CT has taught as a contract English teacher for 13yrs. We've taught together for 5yrs and her English has not improved, there's no excuse. Not once have I seen an English novel on her desk, the only time she'll read or watch anything in English is if she has to.


 

Whilst admittedly dated now, Swan and Smith (2001) give a good (simplistic) outline on how English is/was taught/learnt in Korea, and shows why (traditionally) Korean English teachers hold preference for teacher-centred instruction.

Personally, I think the majority of Korean English Teachers are stuck between a rock and a hard place. They have the knowledge, ability, and willingness to implement a variety of approaches, but exams such as the Suneung treat English as a science, not as an art.
Therefore, the 'real-world uses' for English in Korea is to pass grammar-focused exams.

Even the approach to maths and science, in Korea, is outdated, it's not Constructivist.

I agree that once Korean English teachers get to school, they throw away everything they learned at university and just teach they way they were taught at school, which is a Confucian educational approach.