"You may keep your parts, but not all of them. The barrel, buttstock, forearm, etc. is allowed, without registration or permit. The receiver, and only the receiver, is restricted. If posessed in NYC, it MUST be registered with them.
As you probably already know, the "permit" will set you back one full morning, part of the afternoon, and approximately $235.00. There is no additional "cost" to register your rifles and shotguns, although they MUST be registered."
Could not have said it better. accurate and ON POINT
here is the info for NYC rifle/shotgun permits