I think there's just a culture of lawlessness when it comes to cyclists and pedestrians in NYC. With cyclists it's because we had to ride that way to survive in the past. With pedestrians it's because NYC is a pedestrian town. I don't think you'll ever get pedestrians to stop jaywalking, and frankly, I don't think I want to see that happen in NYC. It's kind of a part of who we are. So it's up to cyclists to accommodate pedestrians and start following the law.
If you want to race through the park, do it early when the pedestrians and tourists aren't around. Many of the road riders already do this, and it would be great of NYPD or the parks dept had officially sanctioned "road bike" hours in the park where they won't ticket red light running. This will of course never happen, but it would be nice.