I'd say the dog park in Fairfield park, a lot of locals use the park daily. Located on Fairfield road at the bottom of Noble Street.
Fairfield Park would be the closest.
94 Fairfield Road
Barbecue (electric), dog off-leash area, picnic area/shelter, playground, toilet, water (bubbler/taps) and shared pathway
Kristy Noble - Noble Properties Brisbane
Fehlburg Park Yeronga, the park also borders on Fairfield