C.T.
Most of the parks in Carrollton do not require a reservation or a fee. You can find a list with pictures, descriptions, and addresses here: http://www.cityofcarrollton.com/index.aspx?page=380
If it doesn't say that it may be reserved, then you don't need to. Its just first come first serve. (I've called and asked about it before.) Just don't pick Cedar Elm park, because I plan to host MY son's party there on the 12th! :)
Most Carrollton parks are not that crowded on the weekends. I know many (not all) of these parks well, so if have a question about a particular one, let me know and I may be able to answer. Good luck!