Hey, don't know if you've found a solution yet but I wanted to drop in.
I'm with you on the whole short hair thing, my father has always been very, 'don't do this', 'you can't do that', 'you're my baby girl' etc. My mother isn't as bad, but not helpful. For me, when I wanted to cut my hair I did it for a charity. Shave for a Cure, to be precise. My parents couldn't argue too much with that without looking like utter assholes and well...despite my father in particular initially being vehemently against (his first response when I told him was 'NO you're not'), he ended up coming to school to support me when it happened, absolute picture of the proud parent.
I know that might not help - controlling parents are hard to deal with - but finding ways to subvert parent's psychology is a good way I've found to help.Turn your biggest critics into your biggest supporters, and baffle them with streamroller bullshit. (basically, don't give them room to disagree, talk over them and BE POSITIVE - talk about how good it is/will be, use words like 'when', don't try to argue with them (that's where they're strongest), but instead assume that it's already decided. I know this won't work for everyone, and geez it's hard, I have struggled over the years, but it gets easier xx)
Failing all that, do what I did last month and cut it yourself
I hope you find a way to get the haircut you want. This is your life and you deserve to live it comfortable in yourself.