I have also seen two species of them in Rome, the rose-ringed and the monks (of which I have some poor quality photos) Climate wise, this is probably less surprising than northern Europe.
Exotic bird markets have long been a thing, so it may be that there have been many small introductions over the last millennia and perhaps further back, I found some references to Ancient Rome. Perhaps only more lately with warmer winters they have been able to establish proper breeding colonies. As an indicator species their movements might be quite telling.
in New Orleans, our introduced parrot is the monk-- same species that is also the introduced parrot of the very different climates of Chicago & Brooklyn, New York! it's interesting to track their march across the planet! in the US, they greatly prefer cities, but i've seen them in rural areas in their native So. America
New Orleans, I bet you get some amazing wildlife in the bayou?
Wow, Chicago and NY, they are obviously far more cold tolerant than I gave them credit for. I'm expecting someone from Alaska along soon!
the Chicago population underwent a large expansion in the 1980s, & i guarantee the winter weather in Chicago in the 80s was famously cold but it seems like if they have enough food (which they can get from feeders from parrot enthusiasts), they'll do well
Harold Washington, first Black mayor of Chicago, in office in the 1980s, considered them his lucky bird & so he & others took them under their wing, so to speak...
(& yes lots of great wildlife, esp birds, in New Orleans)