UPDATE: I have gotten several messages letting me know that this bird is not a thrush at all, but is a brown thrasher. I looked up the brown thrasher, and sure enough, there it is! Huge thanks to everyone who responded, and everyone who cared enough to look at the pictures!
So I went to Central Park and saw several thrushes. This one I managed to get good pictures of, and I cannot identify it. I have been through all of the thrushes and water-thrushes in my references (Audobon and iBird) and none match this bird. Mostly because the eye is yellow. Also the coloring is reddish over the entire back, not just the tail. Please let me know if you know what bird this is. It's maddening! :-)