Dog poop is actually pretty bad for the lawn. As bad or worse for lawns than dog pee.
The nutrient content is too concentrated.
The nitrogen content is too high the same reason dog pee burns the grass.
Let alone the e coli, hookworm and other bacteria.
The reason why parks want people to pick up their dog poop isn't only because it's gross to step in, it's the runoff into the streams and ponds or lakes parks have. Poop from 100 dogs over a 3 day period can effectively close the water for swimming or fishing.
The EPA has even listed dog poop as a dangerous pollutant, even in the same category as hazardous chemicals and oil.