There are several issues to address here:
Most states require parental consent for treatment unless the "minor" is emancipated. If its a 16 or 17 yr. old..( or younger and their condition is life threatening you can treat using implied consent (thats what nursing administration on-call is for anyway).
I imagine that most parents would want to know whats going on (after all, in most instances they pay for the health insurance)
the gray issues are sexually transmitted diseases and abortion, etc. There, you are getting more into "patient confidentiality" as opposed to mom and dad vs. the kid.
So this can be a real ethical dilemma. Heres an interesting, bizarre, legal issue (true story). A 13 year old female comes to the ER complaining od abdominal pain with her mom and stuffed animal, she is 2 months pregnant. The child, because she is pregnant, is allowed to sign for treatment. The mom wants her to have an abortion. The girl wants to have the baby. She winds up having the abortion. Weeks later, she returns to the ER for an unrelated problem- Her mother has to sign for treatment, she cant, because she is no longer pregnant...