JA – First of all, I am super-impressed that you felt comfortable discussing this with your principal! Kol hakavod!
It sounds like your parents don’t want you to switch because they are concerned that the school/type of school you would switch to would be worse for you. I would suggest one of the following:
1. Try to figure out if there an in-between option – a school that both you and your parents can agree on. A school that may be a bit more right- wing than you prefer but is more open than the school you are in now.
2. Show your parents that you really want to grow and that you won’t go otd if you switch schools. Explain to them why you think that switching schools will be better for your spiritual growth and that you think you are less likely to go otd if you switch schools.
Maybe you can show them this by making some kind of deal with them. Like tell them that if they let you switch, you promise to do xyz (xyz being some kind of ruchnius thing, like something in tznius that is hard for you but you think you can commit to and will be willing to commit to if they let you switch).
3. Find a mentor, teacher or neighbor whom you can speak to about this and ask them to speak to your parents.
4. If none of the above are options, stick it out because you have no choice. But try to find a way to make it better. Can you find some kind of outlet – something enjoyable that you can involve yourself in when you’re not in school so that school will be more bearable since it won’t be your entire life? Or can you find something about school that you enjoy and try to focus on that aspect of school?
5. In any case, I think you would benefit a lot from having a mentor.
Hatzlacha! I feel for you.