I got to the park at 69th street and watched the same sparrow war, but right above in the fir tree was the answer. Nest-in-progress with other nests in same tree and lots of available nest-building material on hand, or rather on beak. It's the same as the grackle rookery tree by the path in Strawberry Fields, a nursery tree. Often very noisy.
I'm not sure what this pigeon picked up, but it took a few seconds for the wad to go down.