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My impression of the whole debate over parental notification is that most of those who support are worried about their own children. Parents -- especially parents who are more conservative, forbid their children from ever having sex and warn about the dire consequences that will await them if they do -- are rightly concerned that their children won't share any aspect of their sex lives with their them, including an unplanned pregnancy. So this law is very literally about fear of loss of control of their own families.
Most of these families would probably find acceptable ways to deal with an unplanned pregnancy. In fact, in most of these families, the child would probably tell the parents.
But some people will be very badly hurt by this law, and those people will be the young girls in the worst family situations. The parents who have the healthiest relationships with the children don't need the law, but the parents who are likely to beat, imprision, harshly punish or abandon the child will almost certainly be the ones notified.
Besides, if a sexual predator who is having a relationship with a young girl knows that a legal abortion will lead to his being caught, does that make him more likely to be a good father? More likely to care for the baby and its mother? Or more likely to attempt an unsafe, illegal, life-threatening abortion or just "do away" with the problem (and the girl) altogether?
No matter what the outcome, those in the worst situations will end up being hurt because parents want the added peace of mind and assurance of control. Advocates of the law don't want to admit this, but they really do consider these unfortunate children collateral damage in pursuit of their own security. It's a frightening thought.