Some take Ms. Lynch’s view: They escaped from the airport. Others see less swashbuckling beginnings: Owners sick of their rasping cries set them free. Marc Morrone, who owns Parrots of the World, a pet store in Rockville Centre, on Long Island, says neither explanation is valid. The airport story is an “old wives’ tale,” he said, because shipping containers do not tend to break, and the birds make “wonderful” pets, so why would anyone set them free?

Besides, he said, monk parrots that have lived in the wild are not the same as the pet store variety. “The ones we sell as pets are domesticated,” Mr. Morrone said. “They don’t think like the wild birds do. They are smaller than the wild birds. With a domesticated animal, we control its genes.”

Indeed, the wild parrots, with their acute social skills, have expanded their turf, locally and nationwide, said Dr. John W. Rowden, an ornithologist with the National Audubon Society, who cited data from the organization’s Christmas bird count.

Today in Brooklyn, their pile-of-twig nests are built in the iron gates of Green-Wood Cemetery. They have made homes in Upper Manhattan and amid the trees in Riverside Park. They are in Whitestone and Flushing, Queens. They have built nests in Edgewater, N.J., in the slopes along River Road, an undulating bicycle path in the shadow of the George Washington Bridge, said Corey Finger, of Forest Hills, Queens, a co-owner of the birding blog 10,000 Birds.

But Queens has extended the birds perhaps the biggest welcome.

State Senator Joseph P. Addabbo Jr., who can see one of their nests from the windows of his district office in Howard Beach, is pushing two bills he introduced in 2010 to protect the parrots (also known as parakeets), though neither has passed. One would put them in a protected category. The other would require their nests be handled with care if they have to be moved. The senator says seeing the “green, rather large, rather unique-sounding parakeet” among pigeons fascinates New Yorkers.

Image The nest atop a utility pole in Howard Beach that Elvis fell from. Credit... Ángel Franco/The New York Times

However, not everyone is as enamored of them. In Argentina, “they are reported to be an agricultural pest,” Dr. Pruett-Jones said. In this country, they are considered a “nuisance species,” he said, and have caused economic trouble in Florida, Texas and elsewhere because they build nests around heat-emitting transformers atop utility poles. Several states, including California, have outlawed parrot breeding and importing them as pets.