McAllister Dog Park is in McAllister park. It is a huge people park with a very special place for humans to let their dogs off leash. The dog park portion has ample parking and doggie waste pick up supplies. The enclosed area is partially shaded with seating and a couple of picnic tables. Dogs must have vaccinations/tags. If you google McAllister Dog Park + San Antonio, you can find directions. It is convenient for most travelers as it is close to the airport. Humans who don't have internet access and need directions are welcome to give me a call.
I don't know the exact name of the park but there is a dog park on Old Pearsall Rd. in San Antonio, TX. They have an enclosed area where dog owners can take their dogs off the leash and let them run around. There are ramps and such for the dogs to play on.