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The Smart money is on the cheap, local, good cheese. In Florence that's likely a Peccorino of some kind I think. Parmesan is obviously a Parma thing and that isn't it although I'm sure it's lovely.
Buy whole (small) cheeses if you're flying. Like 1-2kg. Although parmesan is fine at any temp really keeping it in an air tight wrap is not good (that goes for any cheese if it's going travelling). A whole parmesan is 40kg and you need tools n skills to crack (not cut) it :-) A small Peccorino you can carry, keep good and cut.
Mrs IP pops to Florence for work and I ask her to grab a wedge of Parmesan. "The man in the shop said it was Parmesan". Hmmmmm