The parents themselves are using the internet too and possibly are almost as attached to it as the teenagers are nowadays. With our current emphasis on this ever-progressing world, humans want things efficient, fast and easier and the internet provides just that. Its no wonder why parents even though not of the same generation or did not grow up with all these technologies will still use the internet to their advantage in whatever they are doing. Therefore, the parents won't keep the children off the net because they themselves are using it, the only time they will is if the kids are in their way XD
No, I do not believe parents should keep their kids off of the Internet. Completely sheltering kids from something that has become a huge part of society is a bad idea. I do believe, however, that they should be heavily monitored until they are old enough to understand all of the dangers of the Internet.