I don't think its just being able to speak Korean that's getting people more work. I think its being useful and making a good impression. If you can speak Korean it's easier to be useful and make a good impression and you can speak with the principal, vice-principal better. If they like you they will think of you often and want you to participate in extra school stuff. I think being valued and liked might be worth attending some extra things but remember your rights and your contract. If they want you to work out of your contract time or more hours asked to be paid. If its not more then really no problem, why not? I think you would feel better if you spoke Korean but remember if you are too valuable of an employee it can mean more work weather you speak Korean or not.
p.s. If coworkers don't like you very much they probably won't give you extra work or ask you to join in extra activities but do you want to be just average or worse? It's up to you...
Good Luck,
Worya