My host in the Cinque Terre wound up recommending against parking in Levanto, particularly because I was arriving in the afternoon on a Friday. According to my host, Levanto gets packed with Italian weekend vacationers on Friday and parking spots are difficult to find. My host's alternative recommendation was great! After dropping my family and bags at the train station in La Spezia, I drove to a FREE nearby parking lot and walked 10 minutes back to the train station. I repeated the process in reverse when leaving the Cinque Terre. Here's the address of the FREE parking lot, and a link to Google Map directions. Also, the covered parking under the train station was FULL at 3pm Friday afternoon.
Parcheggio Piazza d'Armi
Piazza d'Armi
19123 La Spezia
Italy
https://goo.gl/maps/mu005
Thanks for the helpful responses!