Firefighters say they were called to the home at around noon on Saturday, and a resident told them the kitchen was on fire.
Both people who were in the home made it out, although one needed medical treatment for smoke inhalation.
Battalion Chief Al Magwood says a bird rescue group was operating inside the home and that about 26 parrots died from the smoke.
Magwood says another six birds were brought to an emergency veterinarian.
The cause of the fire is still under investigation.
"Everyone in the bird community is shaken up. There's been a lot of lives lost today," said Robin Horemans, a behaviour specialist with Birdline Parrot Rescue, the group that ran the sanctuary.
"We hope that everyone is praying for their little souls up in Heaven. It's a tragic, tragic, tragic loss and we're all really struggling with it."
The not-for-profit group provides parrot rescue, rehabilitation and support for families that have their own birds, she said.
Horemans said the sanctuary was run out of the executive director's home.
She explained that parrot rescue is similar to dog or cat rescue in that people sometimes fail to anticipate the responsibilities of parrot ownership. She said the birds are highly specialized and not domesticated like dog or cat breeds.
The group's website notes that parrots often outlive their owners, have dangerous biting abilities, and are often very difficult to rehabilitate from broken trust situations.
There also few resources for parrot owners, she said. There are dog trainers, but not many bird trainers.
Horemans said four of the birds that survived were doing well enough by Saturday evening that they were discharged by the vet and were taken to a foster home. Two of the birds, however, remained in critical condition.
"Unfortunately, smoke is very toxic to birds," Horemans said. "Hopefully we will see them in the morning.
She said the group is seeking donations to help with rebuilding costs, or for the care for the surviving birds. The vet costs alone from Saturday were $3,000, she said.