@ShelLuser
OK, installing some sort of malware is inappropriate (and probably a good way to let something bad happen to your home PC), and logging at the router would probably achieve the same ends with better means. But should children have an expectation of privacy, particularly if their using their parents hardware?
I doubt that I'll be able to supervise my children on the net, full time (I already leave them on it unguarded, but at 5 and 3, they've no interest in anything other than the CBeebies page, anyway). I probably wouldn't want to know everything they've posted, but I'd certainly want to know where they'd been - probably the equivalent of them letting me know where they go when their out with friends, but not the detail of their conversations - and I might want to filter certain content (I'm not looking forward to this becoming an issue!).
I get the feeling from cases like this, that it's less a case that the parents want to spy on their kids, but more the case that they want to monitor their activities, but don't really know how.