I think it depends on the laws of your country. If you have the right to own any asset (a house, a computer etc.), then you can own a domain name. You can even register domain names on the name of your dog! The problem is only if there is some dispute over the domain name, such as a court case or UDRP.
The issue of owner is not related to the method of payment. If you pay by credit card, you must be old enough to own a credit card, or get a permission from the credit card owner. But if your parents pay for the name, I think you can even be a baby and register a domain on your name! (or have someone register it on your name).
Google: not obeying their own rules for meta tags "no cache, no index"... please see my personal hompage and Google's cache (linked; and by the way godaddy.com's cheating you too): http://www.speedy.net/uri/en/