There are three in Nashville. Shelby Park, Edwin Warner Park, and West End Park, adjacent to Centennial. It's a matter of taste.
We like Edwin Warner because it's wide open and plenty of parking. The downside is not much shade and not much grass left. It gets muddy when it rains (what's "rain"?) but with the current drought it's more a problem of dust.
Very popular with real "dog people."
West End is smaller with lots of trees and rocks and "interesting things" for dogs to check out. Limited parking. Becoming much more popular since the weather has been so hot. Separate area for small dogs.
Shelby is the original Nashville dog park. It, too, is small and not much parking. It's the farthest away for us so we've only been once.
There is also one in Franklin but I'm not sure where.