No. Oddleberry will only record when (a) it is in the foreground, and (b) you and your child are looking at the picture book section. You can tell it’s recording, as there’s a level indicator that responds to the microphone.
When any iOS app is recording from the microphone in the background, the status bar glows red so that you know that it is recording. This won’t happen when you close Oddleberry, so you can rest assured.