Parental notification laws are laws that require a pregnant minor to notify their parents if she seeks an abortion. This issue is one which tends to win some support from otherwise pro-choice people, particularly parents. All parents want their kid to come to them if they get pregnant. But wanting this and actually being trustworthy are two different things. Abusive parents are a common fear among those who oppose parental notification.
But not all parents are abusive, and many say most are not. But there is still a problem if teens are afraid to talk to them. For right or wrong reasons, teens can be afraid to tell, and this fear may direct them to avoid seeing a doctor entirely. They may try to get a secret abortion from an unqualified, perhaps exploitive individual.
Why do people support notification? The obvious reason is fear for kids. We can reason that we don't know what will happen: will the doctor be trustworthy? How will the kid fare in the procedure? I don't have answers to these questions, at least not yet.
A common talking point is this: "if we require parental notification for aspirin, an ear piercing, an R-rated movie, a field trip...why not a surgical procedure like abortion?" A friend of mine once pondered this question. There's Google search on the keywords "consent aspirin field trip abortion" that will lead you to others who take this point.
Here's one constitutional law site that rebuts it, point by point.