The diamond card is not associated with free parking. Your parking is free if you are staying onsite. What the diamond card does is get you in a possibly closer spot. There is a aaa diamond parking section that is close to the front. We found on more than one occasion there were some spots closer when we hit the park at opening but it we were going in and out at any other part of the day it was definitely very helpful. Even at opening sometimes this portion of the lot was closer than the main section.
Other benefits are your typical aaa benefits for the year towing etc, discounts on other hotel stays you may have, various other things like I found out recently that there is some deal at Tanger Outlet Malls using the AAA card. At Disney world there are discounts on tours - I'm planning on using it for Dolphins in depth saving me I think $20-30, and again for the family magic tour thingy at MK saving another $20 or so.