The two great parks I write about below are in the east part of Bangkok and have a lot less air pollution than parks in central Bangkok. Each park by itself is (I think) bigger than either Lumpini or Benjakitti parks. They're circled in red in this map:
Here's a closer view:
King's Park is fantastic. (Suan Luang Rama IX). Plenty of parking. Fee is 10 baht per person before 5pm (free after 5pm). I don't know what parking costs. There's a nice little dense tree area, ponds, plenty of monitor lizards/turtles/birds, paddle boats, gardens (including a cactus garden, which closes at 5pm I think) and, well, enough for a really nice afternoon. The park is open from 5am to 7pm, seven days a week.
Right next to it (though not connected) is Nong Bon Water Sports Center which has a large reservoir with a long walking/jogging/cycling path that goes around the whole thing. You can even get some kayaking lessons. There's a bike shop - Happy Bike - right at the entrance that rents bikes for 50 baht for 2 hours. They rent out fast on weekends.) I think the park closes at 6pm (might be 6:30pm).
As I mentioned, these two parks are NOT connected, even though they are right next to each other. Something they could easily fix but I guess they're operated by different agencies and each has to have their own little fiefdom, or something. Both parks have bathroom facilities and outdoor public exercise areas/equipment.
One really nice thing about both of these places parks is that they aren't far from a great night market.
Go on a Friday or Saturday afternoon, spend the day at the park, then head to the Srinagarindra Train Night Market (aka Rot Fai Train Night Market) nearby. It's the best night market in Bangkok IMO. (Park on the west side of Kings Park if you want to have a short drive to the night market. Parking on the east side of the park, or at Nong Bong, will make for a long drive as there are no roads that go through. I don't think you can drive through Kings Park from one side to another.)
Oh, and late next year there's a new MRT line opening that will have a stop right by the Paradise and Seacon Square shopping malls - about a 10-15 minute walk to Kings Park. So driving won't be the only way to access this great area.
Hope you'll post here if you go, and share what you think about the parks.
Arrows on this map point to parking lot entrances....