Was there last week and had no difficulty parking in lots in the center of Fira and in some of the smaller villages. I was on the island for a day and a half and regretted renting the car, as it took a while to get it and drop it off (we came and left from the port) and there were several really tight areas to pass through on the way to/from my hotel where I really didn’t enjoy the driving. In retrospect I probably should have just taken the buses (which didn’t look too crowded) or hired a driver for part of the day for all of the hard-to-reach sightseeing.