My husband and I actually checked into this a little when we moved back here. We had a terrific dog park down the street from our house in Oregon, and wondered why there was no such accessibility here.
There are insurance issues, land use issues (parks are for pedestrians and children, apparently, and not dogs), lease issues, and problems with maintenance. Vaccinations and such are also a part of this.
I still think it could be done. Our park in Oregon had good rules that were easy to follow. My son's neighborhood in Cy-Fair has a dog park for small dogs and one for larger dogs, and a place to rinse your dog off after romping. It also has a man-made pool where retrievers (I have three retrievers....two flat-coated retrievers and one golden) and other dogs can jump in, retrieve balls, and do it all over again.

I think it's worth checking into, again.