I agree with Prof. Stairs: even if we fix the argument's conflation of permissions and requirements, the analogies to piercings and 'R'-rated movies aren't close enough to abortion. We need to consider procedures that are of roughly equal invasiveness and seriousness.
So imagine that the 15-year-old daughter needs a tonsillectomy but doesn't want one (maybe she's terrified of even routine surgery, or she's joined a religion that forbids undergoing surgery). Do her parents have the right to force the tonsillectomy on her against her will? I expect that many will answer yes.
Now instead imagine that she's pregnant, and her parents judge that an abortion is in her best interests, but she doesn't want one (maybe she thinks having a baby at age 15 is in her best interests, or she's joined a religion that forbids abortion). Do her parents have the right to force the abortion on her against her will? I expect that many who answered yes to the first question will answer no to this question, including some who say that they regard abortion as morally unproblematic surgery. Your original question is filed under "Abortion," and I think the issue of parental authority has interesting implications for the ethics of abortion in particular.