I was reading this differently, and came up with one, one-word answer:
Ship (or boat)
In whole, you may see me on a farm or on The Moon.
farms ship goods; spaceship
Remove my rear part, you may see me in a prison or in the supermarket.
Prisoners tried (succeeded?) to escape from Alcatraz in a boat by removing the rear wall of their cell; supermarkets have a gravy boat.
Remove my middle part, you may see me on my whole or on your head.
The Hull is a homophone for whole; Seafood restaurant employees often wear a boat hat, with the middle part out for their head.
Remove my middle part again, you may see me on a ship or on a stage.
A big ship uses little ships as tender and/or emergency; stages will often use just, say, the front part of a ship.
Remove my front part, you may see me in the previous word or on the street.
Space-ships, often are launched from rockets that are removed during launch; DC, Pittsburgh, Disney World, and other places do car-boat tours.
Remove my front part again, you may see me on a river or in the theatre.
A car ferry often looks like it has no front part; I saw "Titanic" in the theatre.
Lastly, the title:
The US Navy has parked ships in-between two sides to stop them from fighting.