Yeah, it's worth a conversation with your parents for sure. Maybe you don't need all of that stuff? Have you been sick before? Is there a reason why they are so worried? If they know how much this is bothering you, maybe they will work with you to find a solution. Maybe you could agree on less layers depending on temperature or find an alternate way to school, like getting dropped off or something. If they knew how much it meant to you they maybe they'd be willing to go out of their way to make something else work.
On the other hand... if what you are wearing is totally appropriate, is there any way you could turn it around? Like just say, 'yeah I hate being cold in winter and I'd rather look silly than get cold!' and laugh along? I'm sorry, maybe that's not good advice, I'm not sure. I just know that I purposefully dress like I'm going on a cold weather expedition (bc I hate being cold and I worry about getting stranded somehow) and I rock my giant winter boots all day long... Then again, I am at work, not school and so I get that that makes it different. It's just sometimes when people don't get the desired reaction from teasing they will get bored and stop. If you act like you don't care that might be enough.
I really wish there was a super easy answer. I'd like to think the ppl teasing you are just poking and don't actually know how much it bothers you; like just trying to be goofy/ get a laugh but not straight up mean on purpose.. If that's the case and they are your friends, maybe you could talk to them about it? On the other hand, if you suspect it's totally malicious bc they are just jerks then I wouldn't recommend that. I'd be more inclined to just try as best as you can to ignore them... while keeping in mind that it's absolutely silly and NOTHING for you to feel bad about in any way, shape, or form. I'd like to think that even if they are just being mean - in a few years this will just be a blip that barely gets a second thought from you. And they'll be the ones who feel like idiots looking back once they grow up a bit. And in the meantime you will be warm 😉