I am a law school graduate currently awaiting Bar results. What I offer is mere information, not to be construed as forming an attorney client relationship.
Okay, if you were going 20 miles per hour in a parking lot, you may have been going way too fast. usually, parking lot speed should be somewhere between 1 - 10, depending on type of traffic there (pedestrian and car) etc.
If you live in a state that recognizes contributory negligence, the fact you were negligent by driving too fast may be a complete bar to your recovery by their insurance company. If you live in a comparative negligence state, you recover the portion of damages that were attributable to that driver's fault (so if he was found 20% at fault, he is responsible for 20% of your damages).
Your best bet is to show you were not going fast given the circumstances, and that you were looking where you were going and this person just darted out (and this person admitted to not looking behind him before he backed out).
Also, before you do anything, I suggest reading auto and auto insurance section, and insurance bad faith sections at freeadvice.com's home page.
Hope this helps.