They hired all the actors, they had many of the songs. But what they lacked was a name. According to the article below, the producers of what became The Monkees brainstormed for weeks trying to come up with a name for the Faux Four before the show's network debut. 

They noticed a lot of groups - The Beatles, The Byrds, etc. - seemed to be named after animals or other creatures. So they came up with what they thought was the perfect name: The Parrots. Yes, the Monkees could have been called The Parrots. 

But Davy Jones, for one, didn't want to let this Polly have his cracker. 

"Jones strongly resisted the latter name due to the suggestion that they wear parrots on their shoulders during tapings. “Not me, man. I ain’t having no parrot on me,'” he recalled of his reaction. “Can you imagine? ‘Hey, hey, we’re The Parrots!’ No way.”

Parrots frequently copy human speech. And the band was in effect, copying the Beatles. So it could have worked. But yeah, wearing parrots on their shoulders would have been a deal breaker for me, as well!

Behind the Band Name: The Monkees