Here is information supplied to me courtesy of John Farrel from the Parker Gun Collectors site which helped explain to me what a "Parker" can be....
."Here is some information about the name Parker. In the late 1700s to the mid 1800s the person in charge of the Royal Navy's armoury of ships cannons and supplies was known as the Parker. Ships captains, gunnery officers, and gunnery mates went to the shipyard to fit out their ships with necessary cannon and ammunition to carry out the orders of the Royal Navy High Command while asea. The Parker had a master list of ships and the size of the guns he was authorized to fit them with. He also had a table of what and how much shot, cannon balls, fuse and slow match each ship was authorized to be issued. The Parker was an important position in the Royal Navy. (Source - Patrick O'Brien, author of 22 books on the English navy 1790 - 1815)
The name was also given to the keeper of private parks on estates in the 1500s and later. They were said to have a Parkership, a highly coveted title, and were responsible for arresting or otherwise dealing with poachers. If a poacher killed a deer the Parker was held responsible and could lose his Parkership as a result. (Source - O.E.D.) For that reason the Parker was particularly vicious in dealing with rural folks looking for an easy meal. Mantraps, booby trap shotguns, snares etc. were regularly used to catch and hold poachers for the application of justice.
The common rabbit was called a warrener, parker, sweetheart and hedgehog. They were plentiful in the countryside, and in gentlemen's pleasure grounds. (Source - O.E.D.) Rabbits were fair game for Sunday dinner. It would not be unreasonable to consider that the name on the gun was indicative of what it was intended to harvest - parkers."
