Google is seriously lagging when it comes to updating its YouTube app with all the latest iOS 9 and iPad Pro features. For example, there's no support for Split View side by side multitasking, and the app lacks optimization for the iPad Pro's extra large screen. But what I find to be most egregious, is the lack of support for Picture in Picture mode.
For an app like YouTube, which is solely centered around watching videos, it's frustrating that the app doesn't yet have support for the handy multitasking feature.
But there is a way to get Picture in Picture mode for YouTube videos, if you're willing to use a third-party app. Once such app, entitled YouPlayer, does a very good job of presenting YouTube videos with many of the popular iOS 9 and iPad Pro niceties.