Personally, I believe that parents have every right to try and change the way you dress. I feel that parents should make rules on what their children can and cannot wear, and I think children should respect those rules and their parents wishes. You're basically mooching off them for as long as you live with them, and they're only required by law to provide you with the very basic necessities; you should be very grateful for anything extra that they give you.
However, I do not think that parents should hide their children from their friends. If they allow you to dress a certain way (and if you dress that way they're allowing it), then they should be willing to let those they know see you.