Yes they do.
(...) Alexander the Great brought parrots to Europe from India with specimens of parrots around 327 B.C.
In ancient Rome, pet parrots were considered luxuries by wealthy Romans and they were often housed in cages made from precious metals, tortoise-shell and ivory. Unfortunately, parrots were also considered a delicacy during this time. The Romans introduced parrots to much of the rest of Europe and trade in parrots became a regular business.
In reality, it was not Alexander, but one of his generals.
Besides which, Alexander the Great was not a Roman emperor ...in fact he died some 300 years before the reign of the first person to bear that title, and at the time that Macedonia was in contact with India it would have little or no contact with what it would have regarded as little more than an obscure and minor upstart republic on the Italian peninsula. That's not to say, of course, that the parrots which later graced the homes (and dining tables) of the Roman rich were not ultimately descendants of those original exotic imports to Greece.
This translates to two different English sentences. How do we know which one is intended?
If the sentence were translated, "You with the parrot sing," it would have a notably different meaning from the recommended translation. This is clearer if I use parentheses: "You (with the parrot) sing. I realize these are intended to be simple sentences; so I ask: can the sentence be written more clearly with more advanced Latin?
Tū quī psittacum tenēs cantās: You who are holding the parrot are singing.
Tū cui psittacus est cantās: You who have a parrot are singing (lit., You [to whom there is a parrot--dative of possession] are singing).
I think, as it stands, the Duolingo sentence means "You are singing together with the parrot."
No--you'd be changing the sentence to "The parrot sings with you ," if you wrote: Psittacus tēcum cantat .
"You do it with him" (Tū ... cum eō ) is not the same as "he does it with you" ( tēcum ). Notice the change in verb from 2nd pers. singular cantās ("you sing") to cantat ("he sings"), 3rd pers. singular.
When you say tēcum , the pronoun "you" is the object of the preposition cum, and no longer the subject of the verb cantās . Tū is nominative, but tē , when object of prep. cum , is ablative.