My favorite park is currently Nu’uanu Valley Park. It is located off of the Pali Highway behind Queen Emma’s Summer Palace.
The park has beautiful big trees that provide lots of great shade, even over the playground. There are picnic tables as well as lots of grass to spread a blanket on if you want to have a picnic. The playground has both large and small play structures and a few swings, but they don’t have any toddler swings. There is also a lot of area for kids to run around and burn off their energy. I’m not sure about the weekends, but when I have gone during the week there has always been plenty of parking in their parking lot.