YouTube gives its Premium users who have iPhones and iPads the option to use iOS's picture-in-picture mode. The Google-owned service has only started rolling out the feature, so you might not see it yet. For iOS users in the US, everyone will get access to PiP, but it won't be available for music content. It isn't sure when this option will be available.
To get the feature to work, play a video in the YouTube app and then swipe up the screen to return home. You'll then get a tiny window in the top-right corner with overlaid play/pause and rewind/forward controls.
Source: 9to5Google