As of this writing, Indian citizens with "ordinary" (i.e., not diplomatic or service) passports need an airport transit visa only in the Czech Republic, Germany, and Spain. They do not need an airport transit visa for transit in Italy, as you have found at the page linked in the question. So your family members will not require an airport transit visa for this itinerary if they each have their own Indian passport. Whether or not they have advance parole isn't particularly relevant.
The advance parole document allows them to travel to the US without a visa, so they do not need a US visa for this trip, either. They will need to show their passports and advance parole documents at the US immigration desk, and they will probably be taken to secondary inspection to wait for the officers to check on the status of their pending applications.