Norwood Estate Dog Park
Medium sized dog park, Includes a small-breed area as well as a large-breed area (if you have a little guy/girl that is intimidated by large dogs, small-breeds are certainly allowed in the large-breed area). The dog parents that frequent this park are very friendly. Has plenty of off-leash space including a small bamboo forest that pups love to jet through. Highly recommended.
Red bud isle dog park
Large 13 acre off-leash park surrounded by town lake. Very popular, so parking can be quite annoying, but certainly worth the trouble. It will be your dog's favorite place. Trails are very clean and plenty of water access if your dog is into that sort of thing. One of those dog parks I would visit even if I didn't have dogs with me.
West Austin Neighborhood park
Ideal for south-downtown residents looking for a close place to take the pups. Lake access, wide open grassy fields. Located in Duncan park, so plenty of playgrounds for kids nearby.
Auditorium Shores Dog Park
Iconic park in South Austin. More of a people park if you ask me, but definitely plenty of doggie-amenities and pups running around. If your dog freaks out around bicycles and/or children I'd recommend a different park. Tons of events are held here, lots of outdoor workout equipment, and the SRV statue you've seen in pictures a million times.