As far as I'm aware....you can't. If you play a youtube video (or any video i assume) from the browser, and press the home button or switch to a different app, it forces the video to stop playing. Same goes for the youtube app, if you have a video playing and try to exit it and enter another app or click the home button it simply shuts down the video. If there is a way to get around this, I'm not aware of it. There have been quite a few times where I just wanted to listen to the audio of a youtube video while doing other things on my phone.
You could always try to get the audio off of the youtube video on your PC and then transfrer it to your phone. I use Zamzar - Free online file conversion to rip audios off the youtube if I can't find the audio source anywhere else.