It's pretty simple to add in a video background, so long as you have the video in several different formats and can host the video on a CDN (Content Delivery Network). Setup hosting on GoDaddy or Amazon, then create a FTP and login using an FTP client like Filezilla, Cyberduck or FTPcore.
Then have your videos ready in .mp4, .ogv and a backup image for browsers that cannot display video backgrounds, and embed them like this:
<div class="w-embed video-background">
<video autoplay="" loop="" style="width:100%; height: auto; position:absolute; z-index: -1;">
<source src="http://syddev.com/jquery.videoBG/assets/tunnel_animation.mp4" type="video/mp4">
<source src="http://syddev.com/jquery.videoBG/assets/tunnel_animation.ogv" type="application/ogg">
Here is an example of one I just built using existing video from the web: http://w-video-background.webflow.io/
Feel free to clone it and play around
The parallax effect would just mean you'll need to make the video fixed, and z-index 0.
Getting the videos formatted and hosted will take the longest but is doable with Premiere
Let me know if you have any questions please.