Driving in Cinque Terre

You are right that no visitor cars are allowed in the five Cinque Terre villages of Riomaggiore, Manarola, Corniglia, Monterosso al Mare and Vernazza.
There are small paid parking areas outside each village, listed and detailed on this site (note that things can and do change so information on non-official sites may not be 100% accurate):
https://www.cinqueterre.eu.com/en/parking-cinque-terre
You're visiting at the height of the high season so the chance of finding a space in any of those car parks is imo miniscule to none. Every visitor who is staying ..or just visiting for the day...with a car will be looking for a space.  
A much better option, imo, is to use the train to access the CT villages and leave the car at La Spezia train station car park or in Levanto which has two large car parks. Leave the car when you first arrive and don't use it again until you leave: use the train to explore the villages. There are frequent train services between both places, all the CT villages and to places further to the north and south (e.g. Sestri Levante, Rapallo, Pisa) . Italian trains are perfectly safe and comfortable enough. Use the official Italian railway website in English linked below for train times, details and fares. There's no advantage in buying tickets in advance online for the ordinary Regionale trains you'll mostly be using: just buy them at the station on the day of travel.
 https://www.trenitalia.com/en.html
All those car parks require payment. Don't assume that all ticket machines will accept card payment and always have sufficient cash to hand (notes and coins).
If you are driving a rented car do be aware that any parking (e.g. invalid ticket, parking where it's not allowed) or traffic fines (e.g. entering a ZTL, being caught by an automated speed camera) will be taken from your card even if you live outside Italy, often with additional admin charges and sometimes several weeks/months after the offence.