I've been several times recently to Colonial Oaks Park with my son who is 2. Different days each time. It is shaded and fenced, with only 1 small opening to enter/exit. Each time we go a number of moms and kids seem to show up between 9 and 10 AM. The kids always seem to be between 1 1/2 and maybe 5...
Also, Arlington Park just redid and fenced their playground (green space next to pool). About half of it is shaded and there is new equipment. I haven't been yet, but it looked great when I drove by it the other day.
Forty carrots parenting center is near Arlington Park. They have some programs for mommies and kids, and may be able to point you in the direction of a playgroup.
http://www.fortycarrots.com/extra.cfm?ID=18
The public libraries have some planned activities for your age kids as well.
Selby library has a preschool story time for ages 3 - 5 on Thursdays from 10:30 -11:15 and Gulf Gate has the same age group program on Fridays from 10:00 - 10:30.
Hope this helps. :)