The book Birds: Myth, Lore and Legend by Rachel Warren Chadd and Marianne Taylor states:
[...] The Ancient Romans, who kept both Rose-ringed Parakeets and African Greys (Psittacus erithacus), right, were also fascinated by the birds. "The parrot salutes emperors and pronounces what words she hearth" reported Pliny the Elder.
I found that Pliny himself described a bird that "seems" the rose-ringed (or ring-necked) parakeet (Psittacula krameri). What about the African grey? I couldn't find as easily the evidence for this claim - which, by the way, is not isolated. The Italian Wikipedia page says that "the African greys have been kept as pets for more than 4000 years. Egyptian hieroglyphics clearly show domestic parrots. Ancient Greeks also kept them as pets, a tradition resumed by the Romans".
Do we have reasonable evidence?