We were bike riding in the Cascine a couple of weeks ago, and I heard this distinct, loud bird call. I looked up and thought I saw a parrot - and I think I did, I did, I did see a (wild?) parrot high in a tree. I didn't have tripod but I did have my long lens so I snapped away.
This is the best shot I got. I also did some research, and it appears that there are wild (or feral) parrots in Europe (see here), and in places much colder and further north than Florence.