I agree that I would not chance it with the U-Haul on the parkway, since the Henry Hudson has a toll plaza with a police presence. You an easily take the Deegan to the Cross Westchester, and it's not that much longer of a trip, so that's the route I'd recommend with a U-Haul.
As for parking, that depends upon the street, and what the zone says. Sometimes if there's someone to babysit the vehicle, it may not be an issue with the parking authority, but if you don't have a licensed driver with the vehicle, I'd count on a ticket as part of the cost of the move if it's the only way to load/unload.
All the world's a stage, and all the men and women merely players: they have their exits and their entrances; and one man in his time plays many parts, his acts being seven ages.
~William Shakespeare (As You Like It Act II, Scene VII)
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