It is the responsibility of the parent to monitor their child in any activity they take part in, to make sure they're safe and behaving according to your personal beliefs as a parent. This goes for television, movies, music, extracurricular activities, who they spend time with, and absolutely the Internet. I would much rather have a parent restrict their kid's access than have the government step in for "the sake of the children" and begin to curtail what is and isn't allowed online.
Guarding children is the responsibility of the parent. If a parent chooses to give their child unrestricted access to the 'net, they have a responsibility to ensure that their child is using it according to how the parent deems reasonable.
Having had Internet access since I was roughly 10 (1992), I remember a time when the 'net actually was pretty darn safe for kids. But looking at what's there now... I would be very hesitant to allow a child under the age of 13 or so to have their own PC on the Internet. It would be a family PC in a room like the den that gets a lot of traffic so it can be easily "spot checked" Of course it depends on the maturity of the kid etc... but I think prior to being a teen (and in some cases ESPECIALLY once they become a teen) the parent needs to be very careful.
But ultimately, like in most things, the parent bears the ultimate responsibility to make sure their child is being safe.
Like Baker says, this isn't SPYING, it's BEING A RESPONSIBLE PARENT.