I am not sure the date the Deanna Rose Petting Zoo closes but think it is sometime in October. It is located near 135th and Switzer in Olathe. It is free admission unless you want to pay for the birthday package but would be a nice place to have a birthday party. There are picnic tables outside of the zoo and also some playground areas in the zoo that are great for 3 year olds. It is a great size for little kids. Not too big and overwhelming so they can walk and not get too tired and shouldn't need to be carried and probably wouldn't need a stroller for a 3 year old. Go to google type in Deanna Rose Petting Farm and check out their websites. There are activities in October that cost money so you may want to call and see what dates and times are available that doesn't cost and plan for those times. I think the pumpkin festival was Oct 3-26 and cost 7.00 so you want to make sure you don't plan on those days.
As for free places there really aren't any other suggestions other than a park and the fall can be cold for the park. There are a lot of nice parks in the area though. Antioch park is nice especially for little kids with the small forts and houses to play in but it also has a pond in the middle you would need to keep an eye out as little ones seem to be drawn to water.