Parallax can consume a lot of browser power and is usually disabled by mobile and tablet devices which may not have spare battery power or mobile broadband to cope with it. I believe this is a browser/OS decision. So for example, if your internet provider is throttling back or your Wi Fi is busy coping with too much Wi-Fi traffic, your browser might not show the parallax effect.
The main thing is that when parallax is disabled, that your image is still visible as a static image.
It is for this reason that I never use parallax effects because you can never really be sure what users will see and I don’t like jumpy parallax effects. Ironically, yours looks buttery smooth to me on my system with a fast Wi-Fi in Safari but is visibly slow to fully load. On my iPhone there is no parallax effect as expected but I think it looks just as good.
Your issue could be that you just have too many images which will also strain any browser. Your site is very image heavy and all of your images add up to 2.6Mb when zipped up, which I would say is far too much. With so many images, loading on a slow mobile data mobile could be so long that the user gives up waiting.