I want to determine the approximate total cost of renting a car. One element is parking. Assuming free parking is not available and I must park in a pay lot, what would be the approximate average daily rate for parking with stays in the towns of Arles, Carcassonne, Sarlat, Amboise and Dinan?
You can see Rick's ballpark estimates at https://www.ricksteves.com/travel-tips/transportation/car-rental-costs. More specific details are in the intro to most chapters of his France guidebook.
Easily 25 to 40 euro depending on total time and location. I would budget 50 euro/day.
I stayed at a hotel in Arles where I was able to park for free. In Amboise I had to park in a public lot. I don't remember having to pay for it. We stayed 3 nights in an apartment there. Hotels may have their own parking lots. I visited Carcassonne as a day trip and paid for parking.
You might get to €40 or €50 if you are very unlucky in Paris. I think €50 is way high for the other places you mention. I would expect to pay closer to the range of €15 to €30 a day.
I would be looking for hotels with parking; in my limited experience those tend to run 15 Euro a day or less in smaller towns.