Driving and Parking around Fuji Five Lakes

I will be doing a day-trip from Tokyo to Kawaguchiko in early April. I am thinking of taking the JR train to Otsuki (with my JR Pass) and then renting a car to drive around Kawaguchiko and maybe even go to Fujinomiya in the south side of the mountain. We'll be four people, so the car might even save us money, compared to having to buy Otsuki-Kawaguchiko train tickets (not covered by the JR Pass) and local buses to get around - not that saving money is the main motivation; the idea is to have more flexibility.

My question is this: how easy is to find parking around the points of interest in Kawaguchiko and Fujinomiya? I would really hate to get into a situation where the convenience of having a car ends up being an inconvenience because I can't find parking anywhere.

As a bonus question, is parking very expensive in the smaller towns such as these two?

Parking is plentiful, and generally free. I have never paid in these areas. The larger parking lots at tourist attactions will have parking guides who will direct you to a spot.

Thank you Paul.