Whether it's Myspace, Facebook, Twitter, Instagram, or any other social site, it is the parent's obligation to check what their child is not only producing but seeing on the sites. Kids do need Freedom of expression, but parents need to carefully monitor what is going on with the sites in order to keep their kids safe.
Yes, I believe parents should be checking their child's profile. The Internet is a very dangerous place full of a lot of people who prey on children when they are vulnerable. A parent monitoring the activity on their child's profile could help them quickly catch any of these predators. Parents also would be clued in early on other things, like cyber bullying.
Parents should have be able to check their parents profiles sometimes but not all of the time kids sometimes use it the wrong way but they should be checked once in a while. Some parents do not trust their kids on their private profile and kids don't want their parents to see their private profile