S.H.
Hi M.,
I have a couple of thoughts for you. First, have you considered going to the zoo? They have a petting zoo area for kids with picnic tables, etc. an you can bring in your cake and food.
If you are really wanting a park, I live in North Park by Morley Field which is a great park (it's just east of Balboa Park). It has a playground, plus it has tennis courts, softball fields, archery fields, open space where you can set up an area for soccer, or other fun games for the kids. They have bbq areas as well as picnic tables.