Naxos is a beautiful and very interesting island, full of people during the summer season, where in the center is really hard to find a place to park your car rental. If you manage to find a small place on the street to park your car, be sure that there are no signs showing that it is forbidden.
The best way to feel comfortable in the center of Naxos, is to find a nice private parking place that charges by the hour. If not, you might find a free place on the road, but you will have to pay a parking meter.
Whatever option you think is better for your car rental, have in mind that the parking fines are a little expensive, from 40 to 150 €.