Itinerary advice

These places all sound wonderful, however that is a lot of moving around for a short time. I would drop one of your locations. Count nights instead of days. For example 3 nights only gives you 2 full days in one place. When we went to the Amalfi Coast we flew to Naples and then had 3 nights in Positano, 3 nights in Sorrento and 1 night in Naples before flying home. That way we were also able to see Amalfi and Ravello on the Amalfi Coast and Capri and Pompeii from Sorrento. The scenery is spectacular. Plan what you might like to do each day and then see how much time you will need in each location you choose.

Any longer, we will not spend fewer than 5 days/4 nights in a location. It just seems to work better.

As others have said, this is a lot of moving / driving around between places with slower transportation if you wanted to see both the Amalfi Coast & Sicily. Personally, I'd drop Lake Como. You should be able to fly in & out of Rome continuing on to Naples. So open leg US to Naples then Catania or Palermo back to US. It sounds like this is a 2-week trip on the ground?

You're arriving overnight from a transatlantic flight via someplace connecting to Naples. At that point with elderly parents along, I would suggest it's time to find a hotel at least for one night. Many people base themselves in Sorrento, which you could get to on the same day as arrival. I've overnighted there before going down the AC, that would be do-able but adds an extra night's stay. (I should have mentioned that without jet lag, we spent a week just in Naples, Capri & Amalfi. And the following post from @Valadelphia is even better, cut both Como & Sicily so you have time to enjoy the trip without so many days moving from place to place!)

How are you getting from Naples to Amalfi town? I hope by ferry, not driving those crazy roads in June when traffic will be heavy. I would read up more on the AC, we took the ferry there from Naples in early May & then got a driver to take us back to Naples. Had you thought about taking the ferry? Naples to Amalfi & then Amalfi to Salerno, where you could catch a train (takes longer) or rent a car to Matera? Don't miss Ravenna above Amalfi town, it's AMAZING. You can take the bus, the girls should enjoy the curvy ride.

There's a LOT on the Forum about Sicily, BethFL has done a series of excellent trip reports - https://community.ricksteves.com/travel-forum/trip-reports/14-days-in-sicily-by-bus-train-and-car-in-may-but-with-november-weather

Sounds amazing, but just too much. I would consider saving Sicily for a future trip. You could add the time to the Amalfi Coast area or to Basilicata/Puglia -- both of which are given short shrift -- then it would be perfect.
If you mean three nights in Amalfi, that is just two full days--not nearly enough for the effort it takes to get there. Everyone will be crying to be dragged away after two days! Any interest in Capri or other islands? (sounds like your family might like Ischia). Any interest in Pompeii? Naples is not everyone's cup of tea, but it is a fascinating and exciting city, so I would not discount it without a good read. I would at least add the three Lake Como days to Campania, and I would add the five Sicily days to Puglia. If a Naples return is not what works best, check flights out of Bari and Brindisi (Lufthansa flies out of both). Or, getting to Rome from either would not be difficult. Driving in Basilicata and Puglia is super easy.
Near Matera, there are some mountains towns with a zipline between them--maybe you would be into that? https://www.earthtrekkers.com/castelmezzano-pietrapertosa/
Otherwise, Puglia has cute hilltowns and amazing beaches. Gargano area has the dramatic cliffs, near Gallipoli are sandy beaches with clear seas. More dramatic coastal scenery near Otranto. Lecce is a gorgeous city. Your five days would give you much more bang for your buck here without all the extra travel that eats time.

Five days (really not much more than four because of the long drive from Matera) is totally inadequate for Sicily. So is seven (sorry, Rick). One night in Palermo would be wildly frustrating and two nights on the east coast would mean no time for the Baroque towns of Ragusa, Noto, Modica and Scicli. Furthermore, June can be quite hot down there--as it can in Puglia as well. But one seriously hot area on a trip is better than two. So please hold off on Sicily for a more appropriate time of year when you can give it about two weeks.

For this trip I'd try to have more days in the north than you show on your original plan. Maybe another day at Lake Como and a few days somewhere in the Dolomites like Ortisei. With planning and use of alpine lifts, I think your parents could enjoy the Dolomites.

Your areas if interest, transportation wise, are spread too far apart.
Add a night to Amalfi. 4N.
Ferry from Amalfi-Salerno. Rent car.

Drive to Alberobello, about 3 1/2 hours
You then need to decide if you want to stay there or Matera which needs more time 3N. Closer to Bari.

Fly from Bari to Sicily
Pick either the Palermo or Catania area. 7 nights.

Flight to Naples/Rome

Return home

I posted a similar query about two weeks ago. https://community.ricksteves.com/travel-forum/italy/southern-italy-itinerary-assistance-request . I started with a 3-week tentative plan. My plan is now looking like 30 or 31 days, in part based on the feedback that I received, some of which was very good - take a look. Still, we plan to spend less than 2 weeks in Puglia and less than 2 weeks in Sicily that one commentator suggested as a minimum.

My current thought is 10 or 11 days in Sicily, 9 days in Puglia, 5 days on the Cilento Coast, a couple of days on or near the Amalfi coast and a couple of days in/near Naples, plus a day getting from Sorrento or Salerno to Naples (or vice-versa); I know the train is a short trip, but this "travel day" also includes a leisurely morning in Sorrento or Salerno and nice afternoon/eve in Naples. That evening walk, BTW, is 'la passeggiata.'

For the international travel, open jaw, whether starting in Naples or Palermo and flying out from the other. Both Naples and Palermo have direct connections through central Europe, e.g. Frankfurt and Zurich and then non-stop or one-stop to US west coast.

You could eliminate our Cilento coast piece, add a day to Amalfi and/or Naples and you would still be a day or two under 4 weeks.