We will be driving from France to the Cinque Terre in October. Is there one town where parking is less of a problem?
It's usually easiest to park in La Spezia - near the rail station - and take the trains in. Monterosso might also be an option. Here are a couple of threads on the same subject which may be helpful:
https://community.ricksteves.com/travel-forum/italy/parking-monterosso-al-mare
Since you are coming from France it makes sense to go to Monterosso, the first town you'll hit. Parking in Monterosso is more plentiful, there is a lot right on the beach and some hotels in the new part of town might have places too.
Exit the A12 autostrada (motorway) at Levanto. Follow directions to Levanto then in Levanto at a roundabout take the third exit to the Cinque Terre. After a few km there is a cross roads. Take the right one to Fegina (the new section of Monterosso). Drive to the sea. The last stretch of the road is narrow, so make sure you watch out for upcoming traffic. In a narrow two way road the vehicle going downhill must yield to the vehicle driving uphill. So when you descend to town bear to the right and let the upcoming vehicles go first.