Powell Barnett Park in the Central District is awesome. It's pretty new, and the toys there are big enough to enjoy even if you're a 6'1" adult as myself. Cool park, ok-ish swings.
Location: Martin Luther King Jr. Way & E. Jefferson St.
Number of swings: 2 regular, 1 baby bucket, 1 baby seat
Ground material: Wood chips
Chain length is short.
These swings are noteworthy because they have hard inflexible seats. (side note: "inflexible" was a word I missed in the 6th-grade spelling bee, and it haunts me to this day) It's fun to have a hard swing seat option; your hips don't get any pressure from the sides. Go to this park. Now.
How to get there: Bus routes #3, 8, 84.