PParking in any city is always a problem, but there is a special code in Italy, you should be aware of.
Parking spaces are color coded. Yellow is for the local residents and handicapped spaces. Blue means there is a meter somewhere on the block. Feed the meter and put the receipt on your front dashboard. Warning they do enforce the time limits. White, however is free parking provided there is no sign.
Many cities such as Bolzano, do even allow driving in the central city. Not that you could the crowds are so thick. But there are parking lots very close to the city square.
In Padua, the easiest free parking is on the other side of the river, which can be very close to the Medieval City.
Paid hotel garage parking is generally very expensive, especially they often provide free off street parking for free
In Milan, parking is so difficult, we gave up and used the very good Metro and bus system.