Italy Car Rental FLR or La Spezia?

You don’t need or want a car in CT or Florence
Pick up car as you leave Florence then drop once you head to departure city
The agencies near train station are open on Sunday
As is airport location

could drop in Chiusi (as a small town drop off) on your way Rome or to Florence

Airport choices could be Florence or Pisa ( no direct, few flights) Milan or Rome

Day trips from CT to Parma, Modena etc don’t make a lot of sense
Didn’t look but will definitely be a few train changes

Since you do not need a car in 5 Terre or Florence, I would definitely pick it up when leaving Florence.
I suspect that on a Sunday you might have to pick up at Florence airport anyway, which is far from any ZTL. If you pick up at the train station, the ZTL is much closer but still avoidable...and if you fail, the fine will be less than the extra cost of a longer rental with parking ;)
If your departure city is Rome, you can drive on to Fiumicino airport without issue.
If it is Milan, then I would backtrack to Florence or head to Chiusi and take the train; the drive beyond Bologna is long, traffic-prone, and boring.

I don't believe you'll find day trips to Parma, Modena or Reggio Emila practical from the Cinque Terre, even if you have a car available.

The only place where you need a car is while you stay in Montepulciano. In Florence or Cinque Terre you don’t need one.
Assuming you keep the same sequence of destinations, starting from Cinque Terre, the closest airports are Pisa or Genoa, although neither one has direct flights from the US. Rome or Milan are the only cities with direct flights from the US at the moment. From either to Cinque Terre I recommend taking a train (about 4 hours from either). Day tripping to Parma or Reggio from Cinque Terre makes no sense. If you must see those cities, do so from Florence where you can catch a high speed train, Just add a few night to Florence to take as many day trips as you wish. Florence is a great base for day trips to many places.
You would rent a car when you leave Florence to go to Montepulciano. From Montepulciano the best airport to come home would be Rome Fiumicino (FCO), Italy’s largest airport. You could return the car in Rome (about 2.5 hours from Montepulciano) or in Chiusi (then take train to Rome). If you plan to visit Rome for a couple days before returning home, make sure you return the car upon arrival. A car is not needed in Rome. If you go straight to the airport to fly home without visiting Rome, I would stay at the Airport Hilton the night before the flight. It’s across from Terminal 3 and you can return your car in the parking structure next to it.

I would look at flying into Milan (and possibly out of) and taking an IC to 5terre. You will still need at least one change of trains most likely if you are staying in one of the 5 towns. Depending on which choice you choose, you could also change one more time in Genoa. Runs around 2.5 hrs, but looks easier than what my research from Florence indicates-generally has two train changes and takes around 3 hrs+/- all on Regionale- and since you will be tired……Our trip to 5terre will be from Florence as that is one of our main bases for travel.
We are going to rent a car on our trip and again, research through Autoeurope indicates quite a surcharge for picking up and dropping off in different places, which might be your only choice. We have had very good luck using AAA as a service to book for us through Autoeurope. Adding to that, I’m really trying to figure out how NOT to have a car for any of the time we are in Florence! Means some expensive days on organized tours. Certainly don’t get your car until you leave Florence.
Our point of departure is Seattle and flying into one city and out of another adds quite a chunk to cost of the tickets, but that could be quite different especially if you are East coast based.

Good luck. Have a lovely trip

Airports closest to the Cinque Terre are: Pisa, Genoa, Florence, Milan (Linate is closer to Milan’s central train station). Airports closest to Montepulciano are Perugia, Florence, Pisa, Rome and Bologna.

Thank you all! We will definitely consider your recommendations as we decide which flights to book. RT flights from BOS to MXP are definitely cheaper (as of today, a savings of $600 per couple). The issue is the return to MXP from Montepulciano the day before returning home. We also have a car to drop somewhere (Chiusi, FLR, etc…. Mxp?).
We’ve read in this forum that staying the night in Milan to catch a flight home is tricky if you have a later morning flight and are not staying near the airport. We stayed near Roma Termini (Hotel Aberdeen was perfect for us) the night before our flight home a few years ago, and took the train back to the airport the next morning. It seems that Milan offers a similar option?
It might be worth the extra money to fly home from FCO or FLR.

Thanks again!
Sheila

staying the night in Milan to catch a flight home is tricky

Define "tricky", the airport train from Milano Cadorna station to MXP takes 35 minutes and starts running at 6:30. If Mr. Putin decides to bomb the rails on that very day, you can always take a cab. Dozens wait in front of Cadorna hoping that Mr. Putin will bomb the rails. You'll have to split the € 95 fixed fare, but you won't miss your flight.

You will still need at least one change of trains most likely if you are staying in one of the 5 towns.

The direct IC trains from Milano Centrale call at Monterosso, any other village in the 5 Terre is served only by local trains. Coming from the airport, you'd take the Malpensa Express train to Centrale and the IC train to Monterosso.

Our trip to 5terre will be from Florence

On trenitalia.com try to enter Firenze Campo di Marte ad departure station and an early time of travel.

Means some expensive days on organized tours.

Can't you visit these places by train? Tours often use buses and buses are expensive.

When home airports are indicated, it saves a lot of guessing; so it's BOS? Go to skyscanner.com, search flights from BOS to IT. There is one non-stop, and that is to Rome FCO on Delta. That is not to say there are discounted fares from other sites. However it's wise to book directly with the airline.

If starting at FCO Montepulciano would be next; a 3-hour drive [unwise after the long flight]. I would pick up a Hertz car at the airport, spend the night at the Hotel Tiber in the charming seaside town of Fiumicino [right on the waterfront]; get an early start in the AM.

https://www.wanderingitaly.com/maps/valdorcia.html

Then dump the car at La Spezia upon arrival in the CT; later train to Florence, then Rome [or Fiumicino] the night before the flight.

You mention an interest in "Parma, Reggio, Modena, areas" [also Ravenna]. Maybe stay in Bologna. Florence for a day trip is 38 minutes by train. Longest train ride to any of the towns you mention is 45 minutes.

Thank you all for your feedback. Here's our updated plan for our 15 days in Italy (without all the details). Open jaw tickets: BOS-MIL. Train from Milano Centrale to Monterosso for our stay in Cinque Terre. Train from La Spezia to Florence for our next stay. Train from Florence to Chiusi then taxi or bus to Montepulciano for a week. Pick up rental car midweek in Chiusi to explore Tuscany. Drop off in Chiusi when leaving Montepulciano and train to Rome for our last night. Afternoon flight the next day FCO-BOS. These big decisions are made and now on to the little details!