(CBS) Mira Tweti and Zazu, her pet parrot, are inseparable. Not because Tweti (pronounced Tweety, and yes, that's her real name) wants it that way, but because Zazu demands it, CBS News correspondent Bill Whitaker reports. Tweti explains, "With a parrot there's only one place for everything in your home ... and that's on the floor. He's a huge amount of work."

Parrots are beautiful. They can talk, and do tricks. The average parrot has the intelligence of a 3- to 5-year-old child," Tweti says. With an estimated 40 million parrots in U.S. households, they're not far behind cats and dogs in popularity.

But owners soon learn Polly wants a cracker: hand-prepared food, plus all your time, attention and patience. Mira Tweti should know. She wrote a book on the difficult relations between parrots and people. "They can live to [age] 80," she explains. "They're loud, they poop incessantly, they love to chew. What you've got is a 3-year-old running around with a can opener on its face." And like a toddler, if they don't get the care and attention they need, they bite and scream. Or even worse, isolated in a cage these flock animals go stir crazy, mutilating themselves and plucking out feathers. It's all too much for many people.

Says Tweti: "These are long-lived animals that are designed to live in flocks. The average person would like to have some alone time, the average parrot doesn't want it." Declining in natural habitats, flocks of parrots are multiplying in some places, with 30,000 flying around cities from California to Brooklyn -- set loose by frustrated owners.

Growing even faster are parrot rescue centers, now found in every state. At the Garuda Aviary, part of a Buddhist monastery in Maryland, they're full to overflowing. "So many people realize they've taken on too much. So we get a lot of requests, but sadly, we are not able to fulfill them," says Jetsunma Ahkon Lhamo, the center's spiritual director.

It's such a big problem that the Humane Society now recommends unwanted birds be euthanized. Tweti promised to take care of Zazu when friends couldn't bear it any longer. Every day at some point I think to myself, 'I can't take it another minute'," Tweti says. And every few minutes, someone else is buying another parrot.