The Villages Administration Office at 352-751-3939 says that Dog Parks are the responsibility of the Recreation Department at 352-674-1800.
The Villages allows two domestic animals per home, and yet hasn't provided adequate dog parks.
I would investigate this myself except I have to take my two dogs to Doggie Doo Run Run, a private dog park 3 minutes from The Villages, which is huge, and has grass! Only $25.00 per month, it's a bargain. I think it is at least ten fenced acres with large trees for shade. Oh, and it's a great chance to meet other wonderful dog owners.