I'd like to prototype a web site with a massive use of parallax effect, like this:
For prototyping, I'd like to test how using this effect could influence the navigation and the user experience.
Do you have advice for prototyping it?
Use AxureRP for that. A sample prototype is also available on their forum for that. http://www.axure.com/forum/tips-tricks-examples/8157-parallax-scrolling.html
How to use this effect to better influence the navigation and the user experience?
If you ask that does Parallax has an influence on user-experience then sure, everything you do in your website adds or subtracts from the user-experience but how can parallax be used to do that is too broad to answer. It depends on what is your context and what you are looking to achieve and then you may ask if Parallax can accomplish that.
Build it. That level of animation would be as easy if not easier to build than dealing with prototyping software to emulate it.
I think paper prototyping would be a good fit here.
You could use Adobe Edge Animate with the Edge Commons library. The library includes a parallax effect.