Videopane will allow you to surf and view already selected youtube videos at the same time.
Anything more advanced than that, including non-native video watching, requires something else. Its possible that a couple of tweaks might be in development but nothing 'officially' out as of yet.
If you meant a side-by-side app, there may be one that allows that, but I don't know the name and I don't think it is a jailbroken app either.
There are several AppStore apps with side-by-side video + web capabilities; for example, CineXPlayer (albeit I really don't recommend it as it's a bad player). I've too written some - with free sources, as usual (see THIS).
The multitasking tweak you guys are thinking of is called Quasar. Unfortuantly it's not updated for iOS 7. Matter of fact, I don't think it was ever updated for iOS 6. I wouldn't hold my breath for this. This tweak allowed you to multitask apps. It was pretty cool actually. There was really high demand for an iOS 6 update but it never came.
For just multitasking videos and some other app, like someone already suggested, you can use VideoPane by Ryan Petrich.
^ that tweak couldn't be updated anymore. It was a 16 yr old kid who didn't have the time for it, but there is a tweak being developed for iOS 7 and seems more stable.