I don't think one can expect privacy in a public place; the police can photograph you walking through a public park without a warrant. In fact, anyone has the right to photograph you in a public place.
It seems to me that there is at least an argument that dropping your DNA in a public park is not very different.
You can absolutely expect certain kinds of privacy in a public space. (Like with up-skirt photos, for instance?) The question that needs to be resolved is if collecting DNA violates people's privacy in an unreasonable manner. I think this really falls back on the question of what harm you can do to someone with their DNA.
The EFF apparently feels that you could do quite a bit of harm with it, if not today, then very soon. To me, it seems just about as bad as stalking someone.
It seems to me that there is at least an argument that dropping your DNA in a public park is not very different.