The African Grey parrots, which talk a lot to the family's pet dogs, were taken from a house in Saxilby, Lincolnshire, on Saturday night.
Two of the birds were found by members of the public in the local area.
But a third called Charlie, known for screaming and biting strangers, has not been seen since he was taken from an outbuilding at the house.
Marlene Cavill, whose husband has kept the parrots for 30 years, said they were really grateful to the people who had helped so far on the search but were concerned for Charlie's welfare.
She said the bird could die without medication for his asthma.
One of the returned parrots, Buddy, was found in Saxilby.
She added: "It's great to have him back but unfortunately he has lost his tail and isn't very well at the moment."
Growler, the second parrot, was found by a dog walker in Broxholme, about three miles away.
Mrs Cavill said Charlie may have been taken by two men who were seen "carrying something covered up and it was screaming".
Owner Colin Cavill, 77, added Charlie was known to bite people he did not know and could well have attacked his abductors.
The couple described Charlie as a "real character" who would tell the dogs when it was time for a walk and shout, "you are in trouble", at them.
Police have appealed for anyone with information to contact them.
Two parrots that were stolen have been found but a third "screaming" asthmatic one is still missing.