No matter how clean the parks are, as soon as they fill up with people, I think all bets are off.
Unfortunately a lot of people always try to tough it out when sick and spread the wealth. Honestly I think we have these constant health alerts for the same reason my hair dryer has a label reading "do not use in the shower." Not everyone has good hygiene or common sense. My friend's coworker- an elementary school teacher- showed up to work with the non-swine flu a few days ago.
I'm sure there will be parents there freaking out about sick kids very shortly, if not already.
And of course I do realize other people are contagious before they realize they are sick.