I agree with user14111 that this is possibly "Trouble with Treaties" (1959) by Katherine MacLean and Tom Condit.
An alien (Nifni) ship, the Wllyll'n, meets a human ship the Kemal Ataturk in deep space. The aliens forsee enslaving humanity and adding the Earth to their empire:
The ship, now enlarged to seem larger and clearer, was very alien in design. He visualized alien beings inside, innocently pursuing their course, unaware that their chaotic lives were doomed to be Brought to Order. He remembered the delightful humbleness and obedience of slaves who had fueled their ship at the Thirty-Second World Brought to Order. It made a man feel good, having slaves around. The realization of the sacred Nifni mission to Bring Order out of Chaos spread a warm glow through him.
The human ship is at threat by the Nifni, because it is unarmed and not able to actually fight. The crew don't even have hand weapons.
It differs that the aliens don't succeed in taking the Ataturk; they are bluffed by the human crew, prepared by their telepaths. The humans pretend their ship is called the Vengeance of the Terran Federation Frontier Guard and they belong to a federation of many worlds. The crew's green parrot they claim is an intelligent being (it speaks) of a race called a Wraxtax, the 15th species in their Federation.
The ship's fish are (with the help of a small fake control panel) passed off as atmosphere control experts, the Federation's 25th intelligent species.
The crew's final bluff is that the ship's cat is in charge:
"Now as to the planet for trade center...I would suggest a dead planet of one of the stars near here. It is, of course, Terran, uh, territory, but we would be glad..."
"THE CAT!"
All hands in the engine room came erect and stood respectfully silent. A sleek, black-furred creature, small and walking on all fours, stepped delicately into the compartment, walked about sniffing at the men, climbed to a shelf to look at the viewscreen centered on one of the tubes, ambled about for five minutes or so, then walked out. The men relaxed. One went over and looked at the viewscreen, apparently to be sure everything was all right.
The cat doesn't show any fear of the Nifni, and their leader realizes that the cats are really the rulers of the Federation:
This accounted for the smallness of the ship. It was merely a personal pleasure boat for the Cat, manned by his servants. The Terrans were deceiving themselves with their talk of equality and their "Federation". It was no doubt a device of the Cat, trickery to keep them contented.
Suddenly he understood. The Cats were spreading in a great and growing empire of power more absolute than any Bryllw had ever seen using other species as slaves and keeping them in such hypnotic control that they thought they were free. What need to fear revolt, when the slaves think they are free and are sure they are in charge of their own destiny, merely requesting advice from you?
The Wllyll'n departs at maximum speed to keep the Terran ship from following them to their worlds. And the humans reward their cat:
Chang sprawled on a lounging pad, wiping his face limply. "I never thought we'd work it. You guys could read his reactions but I had to guess. Diu! I'm beat."
"We all are—never played bluff before. We didn't do it too well. The only one with a good pokerface was Shadow."
"THE CAT!" shouted Hahn. They all leapt to their feet, then relaxed.
"Ahh, cut it out, Hahn."
"Give Shadow his bowl of milk or something."
"How come the Nifni don't keep pets?"
"I dunno—how come we do?"
It was originally printed in Star Science Fiction Stories No. 5, reprinted in 1972, which fits your time frame.
