If you agree that these parallels exist, it'd seem that you should conclude that A) Background checks should be required for violent videogames that keep them out of the hands of violent offenders and those diagnosed to have dangerous mental illness and B) There are some games that invite more risk with violence than they can redeem with artistic value.
Don't get me wrong, I hate censorship as much as the next person - Freedom of Expression is a constitutional right in the US, after all. However, so is the right to bear arms.
So, should I favor controlling video games, or should I oppose gun control & background checks?
You may have thought this through, but you've still created an ENORMOUS false equivalence here. Yes there are degrees of violence in games, and there are degrees of deadliness of guns, but games are not tools to hurt others. In fact, violent games help to satisfy one's desire for violence, thereby making the player LESS likely to commit an actually violent act. Given the same mentally ill person, games would help to provide some escapism from those urges while guns would enable the person to commit real violence.
I will absolutely agree that violent games are not an appropriate way to cope with violent urges stemming from mental illness, and such a person should be seeking professional treatment instead, but games are in no way a danger to anyone.