There are several options. I've only parked at either Riomaggiore or Monterosso, but there are options in every town.
Since you come from Montepulciano (Tuscany), we will start from the East.
As you mentioned, there is a parking structure called Park Centro Stazione under the La Spezia Centrale station.
If that one does not interest you because the possibility of break-ins, the next option would be Riomaggiore. To drive to Riomaggiore you have to go through La Spezia (basically along the port). At a certain point there is a fork. Going straight you go to Porto Venere, or you turn right at the fork, and you take the road that goes to Cinque Terre (Riomaggiore, then Manarola).
At Riomaggiore, at the entrance of the village, there is a small parking structure. It's more expensive than La Spezia (3.50€/h or 35€/day) but it's safe. I parked here most often.
As you can see in this photo, when you arrive in Riomaggiore at this sort of roundabout, you will arrive from the road on the left of the photo. You will need to go to the road on the right of the photo (access is limited to residents and car park only), and after about 30 meters you will see this car park on the left of this photo. All those yellowish buildings on the left are a car park. You can park at any level. The first entrance (see the first barrier) is for level 5 and 6, then the second barrier is the entry to level 4, and so on (level 1 is the one closer into town). I don't know if certain levels are reserved for residents, but this is where I usually park for the day (generally coming from Florence). If you go in summer it fills quickly, especially weekends. I don't think you can book in advance and I couldn't find the car park website.
There are also other pay parking stalls in the open, but I've never parked there and they are few. I believe they are all reserved to residents now.
Since you are staying in Manarola, you might even consider Manarola itself. To go to Manarola you take the same road from La Spezia, but instead of turning left down to Riomaggiore, you proceed until the turnout for Manarola a little later. Manarola has its own parking lot (also pay parking lot). I think it's only 3€/h and 20€/day, but it has no surveillance and your car will be out in the elements. Basically it's a little square at the start of the town, that has been converted to a parking lot. This is the photo. I'm not sure if they have time limitations (you'll have to check for yourself). I would go there first, take the luggage to your accommodations, then if you prefer the security of the Riomaggiore car park, you can take your car back there, and take the train to Manarola. I presume the parking lot in Manarola fills even more quickly, given the small size.
I'm not going to bother to describe the parking lots of Corniglia and Vernazza, which are also small like Manarola, but I'll mention Monterosso, since it has two fairly big ones.
One is on the Fegina Beach (see the photo here. It's 25€ a day, but it's in the open as you can see. It has about 300 stalls. The other one is above the old town, it's called Loreto Car Park and, like Riomaggiore, it's a covered parking structure. It has about 100 stalls, and it costs also 25€/day. Here is the photo.
To reach Monterosso coming from Tuscany, you wouldn't exit the freeway at La Spezia, but rather you proceed on the freeway A12 toward Genova (Genoa) then you exit at Levanto. Once you reach Levanto you take the road to the Cinque Terre/Monterosso. At a certain point the road splits. Straight to Fegina, left to Monterosso old town. The train station is near Fegina Park lot.