J.P.
My daughter just turned 2 yrs old. But when she was between 1 1/2 and 2, we took her to Des Peres Park off of Manchester and Ballas. They have two separate play areas - one for smaller kids and one for bigger kids. Then we went to Tilles Park off of McKnight in the Rock Hill area. They have a water play area and my daughter, along with other kids, were just having the time of their lives!! Just make sure you take an extra pair of clothes and towel. CityGarden downtown has a similar water fountain play area for the kids!! CityGarden also has like a wading pool, but we found (after the the fact) that children in diapers aren't supposed to be in there. I saw that someone mentioned Suson Park and I forgot all about that park until just now. That's a good place to go too!! I haven't been there since I was in high school!! You can also try Faust Park out by Chesterfield. They have a huge playground, the Butterfly House is there and they also have an indoor Carousel!! My daughter LOVES that Carousel!!
Have fun!!