L.S.
I live in South Beach and don't really know if any up there or around here. But that does not mean there are not any. We LOVE to go to Coral Reef Par. It is awesome. Lots of trees a lake, long paved trail around to walk or run. There is a playground and lots of ducks you can feed. Our kids love it (2.5 yrs and 6 months). Here's a link: http://www.palmettobay-fl.gov/parks-coral.htm
It is a bit of a drive, but safe to walk around. At least I feel safe.
Good luck.