I don't think parents should read conversations or anything. Any more than they should listen in when their children are talking with friends. They have their right to privacy too. If the parents are genuinely concerned and pay a decent amount of attention to their kids I am sure they are able to tell if something is up.
About the internet in general. I consider myself a fairly seasoned internet user. It takes a fair bit to shock me but it can be done. However, it requires a significant effort on my behalf to actually come across content (usually images) that would disturb the general public. I am genuinely unsure how people stumble across the stuff, maybe a link then a link and so on. I guess the vast expanse of the interwebs can be hours of entertainment for someone new to it, and in the process of exploring they can come across unsavoury content. But it's rare to be truly disgusting stuff.
Of course, it's happened to all of us at some point, I am sure. And for those of us who aren't too old, it probably happened when we were young too. And we are here today debating over it, so it can't have been too adverse We shouldn't try to oversanitise the world for the sake of our children while they are young, they are then only going to discover it all for themselves as they grow and mature, and run the risk of it happening when we are not there to explain and guide them.