Unfortunately, it’s very difficult to get independent or “emancipated” status at your age, and if you live with your parents, it’s impossible. What you CAN do, however, is to write a letter to the beauty school and explain that your parents are not going to help you at all with your tuition bills. Sometimes college officials have a little “wiggle room” when it comes to providing extra aid for students in your situation. I wouldn’t count on it, of course, but it’s worth a shot.
In addition, if your parents’ income is low and, assuming that you are a US citizen, you should qualify for some financial aid. What you might want to do is to apply to more than one beauty school … if there are several options in your area … and see if you get more financial assistance from one school than you do at others.