RIDE THE BUS!
Check out CapMetro & remember, kids always ride free!
While there are some options for free parking on Saturday east of I-35, they are getting thin (try north of 7th St.); there is metered parking all over the place, and the maximum time varies. It’s free on Sunday at metered spots.
Here’s a map of garages and lots, and it lists the times you must pay to park in a metered spot.
Here’s a helpful listing/map of garages with costs.