I see this as a violation of privacy. People say important things through facebook. For instance, I know several people who came out to their friends on facebook. Maybe their parents wouldn't be ok with that. Personal information gets said over facebook messages all the time and it is up to the child to decide when to let the parent know. Parents should educate their children about internet safety and let them learn about responsibility instead of invading privacy and worsening the parent's relationship with the child by being overbearring.
Facebook is a great way for people to stay connected with each other. But on a more serious note, there are young kids posting inappropriate status' or picture, which can get them into trouble in the future. Friends of mine have posted inappropriate pictures of themselves, while the parents believe the kids having nothing inappropriate on their profile. I am not saying parents should read private e-mails. But maybe check who the kids are e-mailing, and see what pictures are up. From the age 15 and younger.
The problem is that I doubt parents would stop there. They would read what the kids posts, their personal messages, and lots of other invasions of privacy. Also, to do what you propose, all parents need to do is be friends with them on facebook. They can then look at all of the kids' posts to other people that are on walls, they can see all the pictures of their child, and look at every friend the child has. If they see some post or picture they deem innappropriate then they can make the child take it down. Also, if they see they are friends with someone who they think is sketchy they can ask the child about it. None of this requires invasion of privacy because any conversations the child wants to do with others can be done in a personal message. Also, if a kid is posting innapropriate pics and things, they are probably doing it on more than just facebook and it is a more serious matter entirely. There is no need for parents to be able to take control of a facebook account.
Some parents may be over invasive with children, but if a child is not hiding anything why should the child care what the parents see? Being friends with your child of facebook may work for some, but in my experience a child with block a parent from seeing posts and pictures, simply by using their privacy settings. Also, most children are not friends with the sketchy people they message. Most of the time (in my experience) the person messaging the child will not be friends with them. Though it may be uncomfortable for the child, it will prevent them from saying/posting something they will later regret.
If the parent is blocked then they will figure this out and have it changed. If the parent is overly invasive when the child isn't doing anything wrong it is even worse. It shows the child that the parent doesn't trust them, thinks less of them, and will just anger the kid. These things put the relationship of the child and parent in danger which has bad long-term effects. If the child is messaging someone they shouldn't and the parents catch them, they will not regret what they did and they will learn nothing. They will only see that the parent is being overprotective and probably continue doing things like that. If the parent is protective and is friends with the child on facebook, screens phone calls, keeps track of where the child is, then the child is not in much danger. If the child does have a bad experience with a sketchy person they have been messaging without the parent knowing, serious or mild, the child will learn what they did is dangerous and never do it again.