Unfortunately, there is no way to remote into a non-jailbroken iPad, as scifan57 points out.
But if you just want to control the movies there is an app for it. iVideoShow
Head over to youtube and have a look at:
and the app is available at the store:
https://itunes.apple.com/de/app/ivideoshow/id501016015?mt=8
It is a tool to start and control presentations (videos) on other iPads.
If you happen to have another iPad - or an iPhone/iPod Touch - you could control your kids iPad.
BUT: Unfortunately, there is a but, you need wifi in your car. Both devices have to be connected to the same wifi network.
It is a little thinking out of the box with additional gadgets - but it can be done. (apart from the fact that you have a wifi network in your car which might harm your and your kid's health. I consider any technical radiation, waves, beams or rays or whatever they are called harmfull.)
Hope this helps.
Knut