Growing up nearby, sometimes my school would have class trips into the city. Being on the buses and seeing the traffic from within was enough for me to decide, "I never want to drive in NYC."
I've only driven inside the city proper twice in my life, and it was as awful as I expected it to be. Trains are the way to go, no question. Stations that go there are abundant in the nearby areas, and the subway can get you almost anywhere you need to within the city, with walking for the rest. Pedestrians dominate crosswalks, traffic lights be damned, like roving herds migrating to the next watering hole. The light turns green for the vehicles, but there's still two dozen people stepping off the curb - traffic waits for them to pass.
I can't imagine living there and owning a car at all. Even if parking weren't an issue, it just doesn't make sense. You need incredible situational awareness, abundant patience, but also the ability to make merges and turns within tight spaces while surrounded by drivers who see your emerging presence as a threat. Both times I've driven there included near-accidents from other drivers.
If you're thinking of visiting NYC, for your safety and sanity, do not drive there. If you're traveling by car from somewhere farther away, do yourself a favor and find long term parking by a train station outside the city. This is not hyperbole, driving in the city is a miserable experience that you do not need. It's one of the few places in the US where pedestrians rule the roads - embrace it and get your steps in, and take the subway for longer distances. Your legs might feel sore by the end, but it's worth it, I promise.