That's fair. I wish I could remember exactly what I did to try and change my embarrassment, or if I did anything at all, so that I could share! If getting sick is the concern of your parents, and you're still able to have conversations with them about this even if they are unsatisfactory, maybe bring up the ways you can get sick by being too bundled up? Like overheating, body pain from the weight of all the layers, etc.?

Or, potentially, bringing a small bag to school where you can stuff some layers on the way to / from school, so that you're not dealing with as many layers at your locker. Not ideal, but maybe better than having to throw the clothes away or pretend to lose them or something like that?