When questions like this get asked about pre-modern cultures, I think to myself: why do we think that medievals were somehow child-like in their view of the world? Do we think that 'science' has somehow revealed some truth to us in the last few hundred years and made us wiser, more insightful, better able to see the world clearly?
Consider the troubadours of the 12th century Occitan world.
Birdsong to the troubadours could be made allegorical, rich with meaning that we moderns can still relate to. In particular, the seasonal changes in the nightingale's (rossinhols) song was a complex foil for many poems, each recognizing the distinctness of the animal's voice and yet hearing a human relateability in the birds' seasonal moods.
Here is Bernart de Ventadorn singing in the 12th c about how much he envies the nightingale's ability to 'verse' in La rossinhols s’esbaudeya ('The nightingale sings happily...'):
The nightingale sings happily
Hard by the blossom on the bough,
And I am taken by such envy
I can’t help but sing any how;
Knowing not what or whom either,
For I love not I, nor another.
Such effort I make that this will prove
Good verse too, though I’m not in love!
And then again, the nightingale is envied for the 'polish' of his song by 12c troubadour Jaufre Rudel in Quan lo rius de la fontana ('When the fountain stream..'):
When the sweet fountain’s stream
Runs clear, as it used to do,
And there the wild-roses blow,
And the nightingale, on the bough,
Turns and polishes, and makes gleam
His sweet song, and refines its flow,
It’s time I polished mine, it would seem.
Oh my love, from a land afar,
My whole heart aches for you;
No cure can I find, for this no
Help but your call, I vow,
With love’s pangs sweetest by far,
In a curtained room or meadow,
Where I and the loved companion are.
In these songs, the nightingale is already imbued with human qualities. And yet, by the allegory the nightingale still stands as an animal apart from humans. And the poems above are in fact not about the nightingale per se, but about love, sadness and loneliness. The voice of the nightingale meant something to the listener, but as rich as the birdsong was to the hearer, they never confused it with humans.
So, with complex poetry like this - perhaps more complicated than most top 40 radio lyrics - do we till think medievals were equivalent to children, infants, primitives; peoples who could not discern human qualities from animal?
For more, see:
Willene B. Clark and Meradith T. McMunn, Beasts and Birds of the Middle Ages: The Bestiary and Its Legacy (University of Pennsylvania Press, 1989)
Joyce E. Salisbury, The Beast Within: Animals in the Middle Ages (Routledge, 2011)
The rose-ringed parakeet (Psittacula krameri) was the only parrot well-known in Europe in the Middle Ages, but even Pliny the Elder wrote about talking parrots in his Naturalis Historia. They were kept as pets, although many native European talking and singing birds were also kept such as thrushes, nightingales and magpies.
Parrots, especially, influenced many works of medieval literature, including the late 14th/early 15th century French Arthurian prose romance Le Chevalier du papegau (The Knight of the Parrot) and John Skelton's satirical poem Speke Parrot.
Interestingly, since parrots were thought to innately say the word "Ave" (Latin for "Hail") by scholars such as Isidore of Seville (an assumption that also appears in later bestiaries) they were later associated with the Virgin Mary and appear in works of art such as Martin Schongauer's Vierge a l'Enfant au perroquet and Jan van Eyck's Virgin and Child with Canon van der Paele. This mystical sort of thinking led to notions that parrots could speak wholly novel sentences, like Thomas of Cantimpré's anecdote in On the Nature of Things that Charlemagne met a group of parrots in Greece who addressed him (in Greek) with "Farewell, Emperor."
Relating to your last question, John Mandeville's Travels (1357-1371) states that parrots can speak "of their own nature," and he ascribes to them a almost human-like capacity for conversation. Conrad Gessner writes in his huge work Historiae animalium that parrots can learn "not only how to speak but to think." Of course, this work was written between 1551 and 1558, so it is not medieval, but he does draw on both Pliny and more contemporary bestiaries, as well.
Sources: Medieval Pets by Kathleen Walker-Meikle, p. 15 and Parrot Culture by Bruce Thomas Boehrer, p. 23-32.