I live pretty close to Overton Park.
Overton Park is great. It's huge, there's a ton of stuff there (trails, a playground, Levitt Shell, etc.), and it's about a 20 minute walk from my house. It's great, but it's not my favorite neighborhood park.
That honor goes to Williamson Park, a small-ish city park that's well hidden between Poplar and Overton Park (the street, not the place). It doesn't have much in the way of features – just a playground, some benches and a long, flat field – but what it lacks in action, it makes up for in charm. It's a quiet. There's hardly ever anyone there, which means that there's plenty of space for playing games, hanging out with your dog, etc. There are no drum circles (at least not that I've come across).
It's a neighborhood secret in the same way that the Target on Colonial is a secret: you either know that it's there or you don't, and once you do, you don't look back.
Where's your favorite park?
Evergreen Historic District
Between Overton Park and Poplar, bordered to the west by Williamson.