I vote with Stephen for multiple bases to avoid excessively long train trips. Although changing cities and hotels is a bit of a hassle, doing it about once a week isn't so bad, and springing for taxis between hotels and train stations removes a lot of the annoyance factor. For a once-in-a-lifetime trip, I'd definitely want to move around some, ideally experiencing evenings in more than one base city.
You might want three bases rather than four, depending on what areas you choose. Even with a month at your disposal you cannot cover the entire country anyway. You can spend a lot of time in the city of Rome itself, and then there are some good day-trips, so you might want to spend more than 1/4 of your time in Rome.
As far as transportation goes, the hill towns of Tuscany are an obvious area where it's advantageous to have a car. A good, comprehensive guidebook to Italy will provide very useful logistical information.
If you're tempted by the idea of renting apartments, take care to be sure the buildings have reliable elevators. That is not something you can rely on in Italy, where so many of the residential buildings are centuries old.
Weather-wise, you'll want to consider your parents' preferences. I may have inherited my hatred of cold from my late mother. I would never have taken her to Italy in April, when it can still be quite chilly and damp. Other older folks have a greater problem with heat.
If your parents have any mobility impairments, give some thought to terrain. The Amalfi Drive area is stunning, but I'm not sure it would a great base for someone physically fragile; there's a lot of up-and-down walking in that sort of coastal area. The same would be true of the Cinque Terre, but folks don't tend to stay as long in the Cinque Terre.
Since you mentioned an interest in religious sites, see if you can include a day-trip to Ravenna, which has a lovely, very walkable (flat) historic district and six or seven sites with magnificent medieval mosaics. They really are heart-stopping. Unfortunately, Ravenna is about a 2-1/2 hour train ride from Florence and even longer from Venice. However, it is a very, very special place.
Assisi has a lovely basilica and is a very atmospheric city in itself, but a lot of the terrain is quite hilly, so the practicality of seeing much besides the basilica might depend on how spry your parents are. Assisi is really a bit too far from both Rome and Florence to use public transportation. A one-day bus tour could be manageable if you don't want to drive--and I wouldn't want to drive out of Rome or Florence.
Much closer to Venice is Padua, which has many worthwhile sights, first among them being the Scrovegni Chapel.