I might have a few things to say.
1. Generally, if you have a long face shape, then it might be best for you to get a short haircut.
2. Why do your parents think that if you have short hair, then you're gay? That doesn't really make much sense to me.
3. Ultimately, the choice is up to you. If you're parents say you can't, remind them that your hair is your hair. You might just be able to get your point across.