Is there a parking zone?

Yes there is paid parking zone as well as free parking zone.The paid parking zone costs for $6 for 1hour parking.

Yes. You have to walk shot through the park. It is close by.

There is parking on both sides of the street in most places of the park that house Japanese tea garden. There is an underground paid parking a short distance away. Underground parking has two sections connected by a 'tunnel', make sure you check both sections if all the spots near the entrance are taken.

Parking on the sides of the road is free. 4 hour time limit I do believe

Hard to find a decent parking around this place. You may have to spend some time around this area driving to find one.

I would recommend parking on MLK and taking the short walk to the garden.