Yes, parents are, in general, both oblivious and apathetic to what their children are doing. Now, nobody get uptight and cry out, "not me!" If you truly do care and are involved with properly bringing up your children, then you are a rarity, to be commended, and you're not part of the problem.
But we can't simply "leave out" the parental component. It is the foundation for how a child learns what is right and wrong. A child develops much of his or her personality core by the age of six. That in itself rules out the education system as the only solution. Parents' first responsibility is to provide for their children - and not just food on the table and a roof over their heads. Parents are the ones who must instill in their children a proper work ethic, a proper respect for their fellow man, a proper respect for authority, and a proper sense of right and wrong. I know some parents who think it's hilarious that their two-year-old has somehow learned to give people the finger and to swear, even though neither parent does those things. They tell me, "They don't even know what it means, and they'll grow out of it." No, they won't! Before long, they'll figure out what it means, and if their parents simply laugh at it now, they are teaching their child not that it's disrespectful and unacceptable, but that it's funny. If parents see their child doing something wrong, no matter how the child learned it or whether the child even knows it's wrong, the parents have a responsibility, if they care at all how their child will grow up, to teach them that it's wrong, why it's wrong, and that there are consequences for doing it. Obviously, if the child didn't realize they were doing something wrong, you can't very well discipline them for it, but once you do tell them there are consequences, then if they continue to do it, you do have to discipline them, otherwise you will be, in effect, teaching them that "I didn't mean any of that, it's really OK, nothing will happen to you."
Sure, the school system can - and should - teach students about the laws of the land, about what is legal and illegal, and how those laws are evolving because of technology opening up new ways of breaking them. But it's that core sense of right and wrong that is paramount. If a child doesn't understand the difference between right and wrong, then they're not going to care whether something is legal or illegal. And parents simply have to start teaching their children about right and wrong right from the start. If they wait till their children are school-aged, they'll be fighting an uphill battle. If they leave it up to the school system entirely, then they'll lose the war. Those few hours in class cannot compete with the constant pressure from popular media, friends, and even video games - pressure to push the boundaries and break the laws, and to not care that it's wrong.