Put usability and accessibility concerns ahead of making the site paralax is the simple tenet that I would advise you follow in order to acheive your goal.
Remember that content comes first, that users need to acheive goals easily. Work on good, accessible navigation and keep content clear and easy to find. Remember users browse on different devices and use different interface tools (mice, keyboards, fingers, screen readers etc.). Think about scroll fatigue, the effects of motion and contrast in colours throughout all stages of any part of the site that moves.
If you have to choose between doing something in a usable, accessible way or doing it paralax, choose the usable, accesible solution.
Paralax is a fad of sorts and many bad sites are produced using this method. This is usually when paralax is put first and usability second, with the wrong choices being made by designers when facing the above question. You will be a better designer and the web will be a better place if you put usability and accessibility first, even if the resulting site hasn't any paralax features at all.
In fact, you can use subtle paralax features to enhance the visual effect, for example, images that gradually move as the page scrolls.
I love Nintendo dearly, but sadly I must share this page and say 'start off by doing none of this': http://www.nintendo.com.au/gamesites/mariokartwii/#gba
Or, in a more positive light, check out the new BBC websites paralax pages:
http://www.bbc.co.uk/news/special/2014/newsspec_7617/index.html - not quite paralax in the classic sense of the word but taking the idea and refining inot something really nice (I haven't studied this link in depth for accessibility, but reference it as a very clear paralax page).
It's also worth taking with a pinch of salt any design award paralax scrolling examples as the people deciding on the awards may fall into the same faddy trap.