My parents never cared about what I watched or played. They only got involved when they decided they also wanted to play; Columns, Zelda, Banjo-Kazooie and Mario were all invaded by my parents deciding they also liked games. I think the worst was the Silent Hill series as they would both gather and watch me play in the dark like it was a movie, they were fascinated.
Though they didn't object to anything I played or watched, they always knew what I was playing and watching, it wasn't like they were oblivious and didn't engage with me about it and if there was something they disagreed with, they would always say 'you know that's wrong' or whatever.
I think problems arise a lot more when parents withdraw from the activities and only learn about things through the media because they get this very distorted view of what their children are playing. A lot of the games the media shouted about my parents would look at and realise that it was really just sensationalist hype and in the end stopped listening. Parents should always take an active interest in what their children are doing, regardless of the activity. If the parent is there then they can say, like mine did, something along the lines of "you know that it's wrong to do *insert moral dilemma here*, its only game and you have to understand that". I completely understand the argument 'I am too busy to get involved' because of whatever reason, but surely any parent can spare one hour just to at least look at what the child is playing, and if they can't even manage that, then that child is going to have more issues from neglect than from video game exposure.
In my experience the only parents I know who ban their children from video games are the ones that have never played one, they just hear one sided studies which were funded by Fox News and see games as devils in disc form. All of them let their children read any book they want, they even brag about how mature their child is because they are reading mature books. I got annoyed and asked a mother once why they allow their child to read slasher novels but not play games; she replied that she would never censor a book because she "wasn't a Nazis besides games are just too violent, they go overboard". It seems in many cases other mediums just have more cultural capital to play with than games and that leaves them open for attack. In a way it's a shame video games weren't around in Hitler's day, if they got burned along with books maybe they would have the same immunity.