The botanic gardens in Ft. Worth is spacious, beautiful and free. I take my girls picnicing and playing out there often. The Japanese garden out there costs (I think)$3 (your little one is free. It is amazingly beatiful and you can feed the fish for 25 cents. Very fun.
The park across the street from the gardens on the Trinity river is awesome too. The have a great playground, lots of open green space and a fun little train ride too.
I also like to take my kids to the Grounds outside the Ft. Worth Art Museums. There are wonderful fountains around the Amon Carter Museum and the Modern has this great iron sculpture outside that you can go inside and it makes the best echos. My 5 year old and 21 month old played in there for at least an hour. Sometimes, when I'm lucky, the kids are calm enough for me to check out an exhibit or two.