Inside Cluny, WALK!
The old part of Cluny is the most interesting, but it's a nightmare to drive there. There's no need to, as there is a good road leading all around it and a large,free parking lot. Inside there are a few parking spaces, but several times I saw the police walking around and putting tickets on the windshields.
The streets are very narrow. One of the ways to go into the old part is through one of the gates from the old city wall,put up long before space was needed for cars.
Try not to drive around in the central part. It is much easier to park your car and walk. There is actually no reason to drive around as everything is in walking distance.