The long history of tense behavior between San Franciscans and their dogs is well-documented. But it's not San Francisco but nearby Foster City that is proposing limiting the number of dogs you can march into a park to three.
While limiting the number of dogs one can tote to a park periodically comes up, often in response to newsmaking events, it has never taken hold in San Francisco. While the notion of licensing dog-walkers and limiting them to seven animals at a time has been broached, no one has come close to walking this one into law.
Per the city's health code, the keeping of four or more dogs (or cats) in one's home may run afoul of the law. But having four dogs at a park? No problem.