I've earned around 250 import parts, not one appears to be compatible with my cars

I started playing late last year and own close to ten cars, some starred up. I've upgraded all to the point they can be, the game displays 0 available parts for further updates for each. I've never used any part — unless I inadvertedly wasted some early on. As mentioned in the title, I've earned roughly quarter of a thousand parts (from the SP, MP and events).

The Gameloft help page is sparse. Is it near-mandatory to sell the less essential parts and buy card packs from the store?

Currently I can't anywhere progress in the career mode (all further chapter 2 events are highlighted in the red) and it's only feasible to drive one car in the multiplayer (the highest stats, most nitro).