Hey. Everyone who have a problem with playing youtube videos in the background in 10.1 and later
Heres a fix,
The next time you open youtube from your safari browser jist request for a desktop site
You can do it either by using the share button in the bottom left corner and say "show desktop site"
Or
You can tap and hold the refresh button on the right side of the URL bar.
Once you open the desktop site
You can now search for your desired video and once it starts playing, just click the home button. After you are at the home screen just open the control centre and you'll be able to play it in the background once you hit the play button in the music section of the control centre.
Peace.