One of the arguments against allowing teenagers to make up their own minds on whether to have an abortion is that it's simply too big a decision for them, and that they're not really competent to make it. And I agree - but given the nature of the decision, it is one that no-one else has any right to make for them.
The decision to have a child is a fundamental question of sovereignty over your own body, and a decision that no-one else has any right to make. Forcing women to have children (or have an abortion, for that matter) is as morally repugnant as forcing them to have sex. In both cases, we must do our utmost to ensure that coercion is minimised.
That's the theory for adults, but what about teenagers? While we have age of consent laws, they're a legal abstraction. We don't ban parents from renting their kids out as prostitutes or forcing them to donate their vital organs against their will because we think rights come into being on your 16th birthday; we ban them because we think that there are some things that not even parents have any right to force their kids to do. It's clear from this that children have body-sovereignty as well. We may limit them from exercising it on the basis of incompetence, but we also deny others - even parents - the right to make those fundamental decisions for them.
Those in doubt should consider the case of a parent who uses their child as a brood-mare for third-parties through IVF or somesuch (it doesn't matter, the point is to seperate out the legal and moral issues surrounding sex). This would attract universal moral condemnation, on the basis that it's grossly invasive, an abuse of power, making a decision that is not a parent's to make etc. But a pregnancy that arises naturally is morally no different from one that arises through technology; if we deny the right of a parent to force a child to continue in one case, we must also deny them that right in the other.
Body-sovereignty having been established, it's simply a matter of how best to protect it. And as I mentioned earlier, given the enormous imbalance of power involved, confidentiality would seem to be obviously justifiable. In order for teenagers to avoid parental coercion in this area and ensure that their decisions are genuinely theirs (or at least as genuinely theirs as it can possibly be, given their age), they must be allowed to keep their decision to have an abortion secret.