This might be more of a drive for you, but it is a great little park. It is called Canterbury Park, and it is on Carvel, between 54th and Kessler in a little neighborhood south of Broadripple. It is just off the monon, so we often ride our bikes there in nice weather. The whole playground is toddler friendly, and there is even a volleyball court made of sand if you want to bring shovels and dig around in that.
I also take my kids to Holliday Park, on Spring Mill at 65th street. They are re-doing their whole playground to have sections for toddlers, older kids, and babies.
Hope this helps.