Prokofiev referred to the bird in Peter and the Wolf as simply 'the little bird'.
"Of course we shall have the flute as a little bird." [†Schlifstein, p.450.]
Although parakeets are smallish - especially the ones we in the UK call 'budgerigars' - as you say, they're not native to Russia.
He doesn't try to imitate the actual sound of any of the animals but to capture their natures: the wolfishness of the wolf, the slyness of the cat etc.
"It must all arise from the concrete, from opposites and impressions: wolf-bird, bad-good, big-small. ... Contrasting characters must have correspondingly contrasting sound-colours." [†Schlifstein, p.480]
IF he had a particular bird in mind I'll wager it was a great tit (Parus major).
Great tits are common in Russia. They are popular subjects in the country's folk art and are mentioned in a proverb equivalent to our "a bird in the hand is worth two in the bush":
“Лучше иметь синицу в руке, чем журавля в небе”
- It is better to have a tit in your hands than a crane in the sky.
I've tried to live by this maxim.
The painter Mikhail Nesterov was a devoutly religious and, in his youth, politically reactionary contemporary of Prokofiev. Prokofiev would have known his work even before meeting him at the mansion of the sugar magnate Pavel Ivanovitch, where he was invited to play the piano at a party. Ivanovitch owned the largest collection of Nesterov's paintings in the world. He had no interest in the avant garde, but Nesterov, who advocated simplicity in life and in religion, produced colourful and folk-inspired work which was right up his prospekt.
His paintings often include great tits. Holy Rus for example, has three or four of them lodged in the silver birches. His use of them is decorative: similar to their use in Russian folk art.
So Prokofiev would have known this colourful bird. And although it is best known for its repetitive "tee-cher tee-cher" call, an individual bird can have up to eight song-types. [ecologicablog] So perhaps Prokofiev heard a particular phrase...
† S. Prokofiev. Autobiography, articles, reminiscences by Sergey Prokofiev; S Shlifstein