We may be allowed to re-seed the dirt, and get some turf back! However, this would require closing that area of the park for a couple months, with the south end where there's still grass remaining open. Additionally, this will cost about $5,000.
The others options are to leave it dirt, or use a cheaper material like straw. Apparenty Harry Griffin dog park has used straw successfully, though it takes some volunteers willing to spread it around regularly.
So if turf is what we want, donations are needed! And if not, please provide some feedback.