My parents are like that too, but they've kind of convinced me to not be so trusting. I once chatted with this guy and I felt I had a connection with him, so I felt really close to him. My parents didn't like that one bit and eventually the guy cut off contact with me. I thought he was decent enough though, not a bad guy, but I haven't heard from him since.
My parents are right though, that I do need to be cautious. I have a myspace and all, but I will probably never meet with a friend who asked me to meet with them because you never know. So, yeah my parents have made me a little paranoid perhaps, but I just can't take the risk. It's kind of unfortunate because if I met someone I liked online I would never get to know what they're like offline. Too dangerous.