Try to get the rule changed so it reads "no UNLEASHED dogs allowed on.... " And try to get them to add some kind of punishment for those who don't clean up after their dogs (i.e. a $50 fine or something like that). Sure people can bash HOAs (and I totally agree with why they are hated so much), but the rule was probably not enacted because of one incident. It was probably put into place after many issues with owners not cleaning up after their dogs, so they felt they had to do something.
As with all HOA issues, the solution is to be in on it.... be part of the HOA. Run for office and get on the board. Being part of the process makes it that much easier to control what rules get put up.