People have been tooling around Golden Gate Park in their cars for more than 100 years, but it’s never been tougher than now to get around and find a space on the weekends.
This is what MLK Drive looks like from 9th Avenue on a summer weekend – it’s a parking lot. You’d be better off walking:

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And here’s where you’ll find cyclists and bikers on MLK when things are bad – they just lane-split in the middle of the road. That’s how bad it can get.

But you want to carry your brood to the de Young Museum or the California Academy of Sciences anyway, right?
Now, if you want to approach the park from the north, from the Richmond District and then head right to that $4 per hour Music Concourse garage, well then be my guest. The problem with that is that the garage gets full sometimes. In that case, you should be able to (eventually) find a space on Fulton (for free on Sundays and some holidays.)
Or you could make a try for Stow Lake, where it’s always free to park. Pretty easy to get to from the south. Just follow the blue arrows past the boat house - if you can find a place near the east end of the lake then you’ll have to walk just a couple minutes to get to wherever you’re going.

Whatever you do, don’t try to circle the Music Concourse or use MLK between 9th Ave. and the southern entrance to Stow Lake. You’ll be traveling at one MPH if you’re lucky and there’s a good chance the southern garage entrance will be blocked off. You’d be better off trying to find a free space on MLK east of 7th Ave. or on Lincoln (where it’s free to park on Sundays.)
Bon courage.