Although the scrolling was not working quite right even on the link, I can advise this ability is not in Striking, and realistically, I believe it would be a major undertaking to add it. It would require modding the page template file you choose to work with, plus the themegenerator file, functions file, and possibly 3 or 4 more other files, etc. I have never examined parallex scrolling in any detail although i did look at some of the jquery for it some time ago, and I don't believe KaptainLin has either. So on our side we don't have the familiarity in order to point you in the right direction based on its needs.
I think you would have to find a php programmer with some time on their hands to work the integration.
Just to clarify I was referring more to the smooth scrolling element. I know parallax does some fancy things with the background but after reading some more I found out that it was called Smooth Scrolling. I want to make anchor tags for my nav menu and have it jump to each section but instead of jump just scroll down.