Parma - Emilia Romanga exploration

Lonely Planet covers the Emilia-Romagna region which is a pretty good guidebook: https://shop.lonelyplanet.com/products/italy-14?via=Z2lkOi8vbG9uZWx5LXBsYW5ldC9Xb3JrYXJlYTo6Q2F0YWxvZzo6Q2F0ZWdvcnkvNTllZTQ5YjlmOTJlYTE0MTg2MDU4NThm.

You'll need a lot more than one car. Even a van may need more than one. Luggage will be the deciding factor. I would google some foodie tours of the area for ideas of itineraries. They might also offer an option for private tours given your numbers, or plans you can replicate yourself.
Alternate to cars are trains. I am not as versed on current Italy pricing, but Spain, for example, offers lower costs for purchasing a 4 seat block of tickets. They would be the 2 facing sets of seats sometimes with a table in the middle. Check the train routes from Parma to the other cities of interest for convenience.
Even if you don't think it would be interesting, do consider the local car making factories and museums. Even me, a senior woman, would be thrilled to test drive a Lambourgini or Ferari while in the area.

I just did a week in Emilia-Romagna last month and I decided to base myself in Bologna. That was because I wanted to be centrally located in the biggest transportation hub. Parma is certainly more atmospheric than Bologna. I think you should be able to do everything you want with trains, but it depends if you want to venture out into the countryside or not. With such a large group, not having a car might actually be easier.

We based in Bologna three years ago. We visited Ferrara, Parma, Modena, Ravenna and San Marino, all by train or bus. But Parma would be good base as well, smaller town than Bologna.

If you want to move almost daily for visits the central cities of Emilia-Romagna are the bette hub for your group: Parma, Reggio Emilia, Modena or Bologna. Bologna is better for moving by public transport, mainly by train.
For food tours anyway is better having your own car (or a little bus), because the most of places are far from the center of cities and towns and usually hard to be reached by public transport.
3/4 days could be enough, but you can stay even a whole week if you want to visit both food producers and a little of art and historical places. Imagine that you can dedicate a day per every province.
As suggestion I can say to contact Italygate, a little local tour operator located in Modena (even if the website is temporary down for a restyling).

In 2019, we based out of Bologna because of the good train schedules to the various towns in the region. We took a day trip to Ravenna, spent a half day in Modena wandering around and spent a couple of days exploring Bologna. We also took a food and wine tour with Italian Days. We were picked up at our hotel at 7 AM and headed to the cheese “factory”. We arrived in time to see a great demo on how they make Parmesano Reggiano from start to finish. Our next stop was at a place where they make balsamic vinegar. They explained the process and we had the opportunity to taste several different ones. Our 3rd stop was at a prosciutto production place. We then proceeded to lunch (around 3 PM) and were served a proper Italian meal. The food just kept coming and coming! We had three different pasta dishes before the secondi and dolce! Plus all the prosecco and wine you would want. This was a very nice tour and I highly recommend it.

Thank you for your replies and information. We have decided to base out of Bologna ... couldn't find a hotel in Parma that met our needs. Lots more research to do, but fun to have your ideas. Thanks!