My parents have been selectively supportive of videogames. That is to say they do accept them as a popular form of entertainment but find some things a bit too crazy for their liking. For example my mother really doesn't approve of violence, like most mothers don't and finds needless violence to be one of the drawbacks of videogames, for example Manhunt would disgust her.
However my mother definitely approves of games like Monkey Island and Portal, puzzle games in general really, for stimulating brain activity. She also liked the Mega Drive era of gaming and insists that was the best games system we had with games like Sonic and Lion King, simple in their vision, brilliantly realized and reasonably challenging platformers that weren't violent. I kind of agree in saying that the mega drive was one of the best consoles I've ever had.
Moving onto my dad, he's been more hands on with videogames since I was a child, I'd say since the PS2 onwards though he's been detatched from gaming, suffice to say age has caught up with him and he no longer has the reflexes to play fast paced titles.
My dad introduced us to gaming by taking us to arcades as children and letting us play Beast Busters and the X men arcade etc. My dad also did all the research before buying us a games system, he was also a great guy for getting involved as I played Lion King, that game had a steep difficulty/learning curve and he was there to make sure I didn't give up, so yeah pretty swell dad when I think about it.
His most recent gaming experience before he kinda dropped out was the Ace Combat series on PS1, he really got into those games and often played them with us.
Anyway sorry for the wall of text but it's a good topic and I like to see how parents use videogames to connect with their children and help their growth.