Elinor Friend Playground at the San Francisco Zoo
The Elinor Friend Playground at the San Francisco Zoo is beloved by San Francisco parents and children. The layout of the play areas separates children into 3 distinct biome based playzones each appropriate for a different age group. The result is that you don't have the big kids trampling little kids problem that exists at many playgrounds.
There's a zoo cafe right at the play area making lunch and picnicking a breeze. Don't pay for zoo parking - there's plenty of street parking on Sloat Blvd.
The zoo does host free zoo days which they announce via their calendar: https://www.sfzoo.org/calendar/