Prevent apps from pausing Google Play Music
The behavior is determined by Play Music, which gets notified when another app requests audio focus (as per the documentation).  Unless there is an option in Play Music's settings to disable this behavior, or perhaps a mute function in the "problem" app, there's nothing you can do about it.

It's also possible for other apps to pause by sending an ACTION_MEDIA_BUTTON intent, which is not protected &mdash; which again means there's nothing you can do about it unless those apps have an option to not send it.