At some point they did and then they kinda stopped caring once I got to an age where you're not really at risk for being targeted by child predators. They continually berate my online friendships though. As a matter of fact they kinda go out of their way to sabotage them, so I generally keep it under wraps.
As a matter of fact I think I can pretty much blame my parents for every stuffty experience I've had regarding online friends past like, the early days of internet handicapation. The support of the people you live with can really make or break that stuff.