How are minor league hockey teams financed? Does the team pay the players themselves, or are their contracts paid for by the NHL parent franchise? I recall that before the 1967 NHL expansion, Hall-of-Famer Eddie Shore owned the Springfield Indians AHL team and owned the contracts of its players. He often made injured players earn their keep when missing games by selling programs. All players had to help clean the arena. The LA Kings wanted one of Shore's players, but he refused to sell. Kings owner Jack Kent Cooke bought the Indians from Shore to get the player.
A player may be contracted by the NHL parent franchise and move between the clubs as you would see in the MLB, or the player may be on contract with the AHL team specifically. A third scenario also exists where they could be paid for in some combination by the NHL and AHL teams. There are different NHL contracts that allow for different situations:
Most players are on NHL contracts because the AHL is used as a minor league system. They are often young players with hopes of becoming NHL players, or older players that could play a role on an NHL team in the case of an injury.
Each AHL team can be one of three things:
The contracts, affiliations, and ownership also apply to the ECHL and CHL.
The situation that you describe would most likely refer to a privately owned AHL team that had the player in reference under AHL contract. The rules today may be different than in 1967. Today if the player was under NHL contract the AHL or other minor league team would have no say in the transfer to the NHL club.
Lastly as for the franchise finances:
If the AHL team is owned by an NHL team, they would be free from worry about operating expenses or player contracts. This would all be the responsibility of the NHL team. This allows for more control over coaching and playing time from the NHL perspective. If they want certain players to play certain positions or minutes then it is passed down to the AHL team.
If the AHL team is independently owned, they will be in charge of the operating expenses and the non NHL contracted players on the roster. The AHL team may hire a coach they see fit and player the players the way they want.
There are obvious upsides and downsides for both the AHL and NHL clubs in each situation.
Here are references used in the answer: