The Miller Playfield on Capitol Hill is another great park. It's next to a school, and is quite often full of young children during the day (but what playground isn't?).
Location: 19th Ave E & E Thomas St.
Number of swings: 5 regular, with 1 baby seat swing at the end.
Ground material: Wood chips
Chain length is medium.
The great thing about this playground is that there are more swings than usual. Most Seattle parks have sets of 4, usually 1-2 taken by baby options. Having 5 regular swings in a row at this park gives you a better chance of getting a turn.
There's also a tire swing here:
How to get there: #43 Bus