Apple - Record video while playing TV on Jio Phone
Check out Record and Listen. Jamcam is great, but only records videos for up to 15 seconds.
I helped develop this iPhone app just for this purpose. It is a video recorder that won't pause your music.
An app called "Jamcam" is the only way I've found to do it, might be better options out there but I don't know them, hope his helped (: