If you've read much about dog parks, you will find that the most important word is "park." Dog parks usually exist as part of, or adjacent to, a people park. There is usually a playground, often a basketball court, and may even be a baseball diamond. Frequently, there's at least one tennis court. There are walking paths and maybe an open field for soccer or other games. Sometimes, there's even a swimming pool. Odds are that your dog isn't welcome in these parts of the park.
It can be difficult to find a dog park that puts the dog first. But they exist. Consider Red Bud Isle Park in Austin, Texas. The park, which is actually a spit of land sticking out in Town Lake, is thirteen acres of puppy paradise, especially for water breeds and other dogs that enjoy swimming.
Red Bud Isle doesn't have any fences. The surrounding water provides a natural barrier. Other than unpaved trails and a few poop bag dispensers, the park doesn't have anything for people. It is a dog park, after all, and all they need to have fun is open space and maybe other dogs to play with. And you will find plenty of those at Red Bud Isle.
It's a very popular dog park, in part because the park is well-maintained by a veritable army of city workers and volunteers. You won't find a lot of trash or debris on Red Bud Isle, even though it's in Austin. Unfortunately, you won't find a lot of parking, either. The park was created before the city realized just how popular it would be. However, there are other options in the surrounding neighborhood, if you and your dog are willing to walk a bit.
Red Bud Isle is not a real island, but it is surrounded water. However, it's important to realize that there isn't a beach. The shore surrounding the park is rocky and has downed trees, so if you are going to play in the water with your dog, it's a good idea to wear shoes that can protect your feet and provide some traction on slippery rocks.