Thank you guys!
When I convince my parents... I'll start

There is a few easy ways into this.  First, I would not recommend labeling the diet, like "Paleo" or "Zero-Carb" etc.  Just tell your parents you would like to cut out all sugar (in all its various forms) for your acne problem.  The next week, tell them you would like to start eliminating all grains, tell them you saw a Dr. on television or something.  Next, tell them you think you are highly sensitive to chemicals and refuse to eat anything out of a box or other packaging.  Some might argue, but if you can get to eating real, unprocessed, chemical free, real food (excluding grains and sugar of course), you will be well on your way to a healthier way of eating, and you might see some benefit in your acne and hair as a result.

Then, if you so choose, you can move on to the more finer points of Paleo or whatever WOE you choose.