Parents naturally have a tendency to think of the worst that can happen to their child, and thus become over protective when something has the potential to harm their child.
Parents hear horror stories of the internet all the time.
My mother was once watching a show about little girls who were lured by online predators, and the horrid things that happened to the girls.
As a result, the parents assume that if it can happen to her, it can happen to their precious little angel. They assume that the children are too stupid to fend for themselves on the internet, and keep predators away even though in my three years as a teenager on the internet, I've never been tricked by one. They believe that their children will be lured, and agree to meet anyone they see online, even a pedophile.
However, I can't blame them. Children tend to be idiots, and it's happened before. they've been kidnapped by people they met online and decided were okay without their parents' consent.
Parents also aren't tech-savvy. They don't realize the magical ability of being able to monitor your child's behavior online. They don't know about setting private profiles, or checking your kid's history, or just knowing what they're doing. Parents don't know they can, so they don't think it's possible.
They think that precautions mean nothing because of what they've heard about kids being kidnapped- even though every pedophile story involved stupidity of children and lazy, uninvolved parents.
They don't take the chance because they don't think they can reduce the risk.
Also, 12-year-olds aren't allowed to register for Myspace and Facebook if I'm right.