What about a "dog run" as opposed to a full-on dog park? I've seen these in some cities, and they are generally well used, well taken care of spaces that create a hub of activity without making the area feel overrun by dogs.
Sort of like this:
or this:
Source: dogspin.com
You could probably cram something like either of these into Juba Park or the patch of grass beyond the right field fence at Shaw Park without too much trouble... it wouldn't have to take up that much space.