not sure my solution is the best, but:
You set your Background layer parallax to (0, 0)
You createanother layer for your button's levels, with a parallax of (100, 0), to make this layer scroll on the X axis
and You use the scrolling to make it scroll (a ScrollTo behaviour on a sprite maybe, and you move the sprite when necessary)
And that is why you shall respect the bug reports guidelines, not only giving a capx is making the bug reproductible in one click in a situation they can work with (less time wasted trying to reproduce vague instructions) but also it helps filtering false positives.
Game design is all about decomposing the core of your game so it becomes simple instructions.