Welcome to the forum. Recommend you check out the port of call section for lots of good info on things to do and places to see.
Jamaica:
https://boards.cruisecritic.com/forum/67-jamaica/
Cayman Islands:
https://boards.cruisecritic.com/forum/89-cayman-islands/
Enjoy your cruise.
This should be helpful:
Contact Us
For inquiries regarding Manhattan Cruise Terminal parking, email Jason.Rubino@propark.com or call 646.299.4670.
https://nycruise.com/manhattan-terminal/parking/
Recommend asking/researching on the Florida Departures forum where this thread will most likely get moved to. Link:
https://boards.cruisecritic.com/forum/386-florida-departures/