I wore similar clothes--a puffy parka that went down to my knees with a furry hood, boots, a hat, scarf, oversized boots! It didn't help that almost all of these things were a size too big for me, haha. But yes, that makes sense, getting all the layers on/off can feel like an ordeal. That being said I feel like a lot of other people have experienced embarrassment from overprotective parents too, even if in other ways, so they probably understand that it's your parents and not you who decided on those clothes!

Would it help to think about what people who care about you for your personality think about you, rather than those who you might worry about seeing you? They probably like you for reasons other than how you dress, and who knows maybe they even appreciate it as a quirk!