Hello J.A., giradman.
Yes, sorry to have omitted those details (in fact we have two, and the same problem occurs on both iPad Air and the previous model, name of which escapes me). Both running IOS 8 or whatever latest is. (Yes, I do connect by wireless of course).
Anyway, having thrown in my pebble and disturbed the water, I've now also solved the problem.
The trick is Auto Lock, which normally puts the iPad to black screen after a set time. Go to Settings on the iPad and change Auto Lock to 'Never' from the more usual 2 minutes. Then iTunes (for example) stays up and live on your iPad screen ad infinitum, or rather (disadvantage) until the battery runs down.
Answer of course is to run on mains charger if you're connected to Apple TV for a long period.
Oh, and don't forget to change Lock back to 2 or however many minutes, otherwise screen stays 'live'.
Overall benefit of this: if you've got a good stereo system attached to your television, you can play your iTunes music through it, via an Apple TV, and enjoy the benefits of 'proper' amplifiers and speakers. Airparrot will do the same job on a Mac computer, i.e. it will MIRROR on your TV whatever you've got on your Mac screen, but I don't think there's a similar app available for iPad.
Allen.