Tell your co-teacher that they have to be the one to explain to the students that they have changed their mind and won't allow the movies.  Make them explain exactly, that it wasn't you and you aren't changing your mind.  Otherwise you are going to loose the trust of your kids and they aren't going to believe a single word coming out of your mouth.  And it may be horrible of me, but I hope your kids cry and whine and milk it loudly and the Korean teacher has to take all of it.

Also - try showing them the materials that you have created to use after the films have been completed.  Maybe they will change their minds and let you continue on as planned.

Really though, you need to have a sit down with your co teacher and explain how detrimental it is to your process to have to go back on this harmless promise that you had only made AFTER having it okayed by them.  Make a point of saying how unprofessional it is (this tends to get my school to give in when I say it - and I only say it when they really are).

As for other systems - I have a money system.  I give out fake money in class (also fine them when they are behaving poorly) and then have a camp shop later where they can buy snacks and stationary (and my kids can pool their money together for a movie..).  You can get it buy people buy using it to teach shop lingo (how much? discount? that kind of thing).

I don't know why anybody bothers with reward systems anyway. They aren't worth the effort and time they cost to create and keep track of. If you make your lessons interesting enough, and present them with enough enthusiasm then the kids won't need any external reward to keep them paying attention. The lesson itself will be the reward.

The parents complained.
It's not really your co teachers fault though I guess they didn't argue your case very well but then they might not have had the chance. The complaint might have been made to the VP or such, then they chew out your co teachers and then the co teachers had to break it to you.

Throwing it back at them is not the best way to handle this.

Let the kids know it was because parents complained. Sorry Kids.... then move on.

Plan some games involving English (like Mafia/Werewolf). If the school is more strict about the kids learning then you should make a word search and other puzzles involving vocabulary they need to learn.
A full on puzzle day would go down well.

I know you are disappointed. You are trying to implement best practise in the classroom but got thwarted by the system.
You have to move on though.

Replace the movie reward with another reward that the students would equally enjoy.

Make sure your remaining classes are fun and they don't 'feel' like they are studying, so they'll forget about being able to sit and watch a movie.

Not sure if this would work, but what about chopping the movie up into clips and using bits at a time for your lessons instead of a whole class devoted to a movie? It's essentially the same thing in the long run, but this way you get to throw in segments of both movies and English practice. If the kids start asking questions, tell em the parents complained about a full movie class.

 I personally am happy lots of people are being discoveraged from shoving movies .  I think it is a complete cop out and defeats the purpose and the expence of hiring native speakers.

If you really have to show movies at the very least show them with English subtitles.

...... only to back pedal now that parents have complained......

Are you at a public school? I doubt parents complained and this is just being used as an excuse to make the decision seem legit.

Your main problem seems to be timing. A movie day this time of the school year isn't really appropriate. I'm guessing your co-teacher or VP doesn't want to just come out and say it. Maybe you could delay watching the movie until after the students have finished their final exams. There is always a week or two to kill with no curriculum and classes full of disinterested restless kids.

I'd double down on your reward system and say that the students will get some snacks or drinks to enjoy with the movie. (If you pass it by your co-teacher again)

I personally am happy lots of people are being discoveraged from shoving movies .  I think it is a complete cop out and defeats the purpose and the expence of hiring native speakers.

Meh... Have your worked public school in Korea? The last two weeks at the end of every semester turns into a mini film festival in the students regular classes. The OP is to be commended for limiting it to one day and using it as a reward instead of a time waster.