To obtain new spare parts, you need to open Supply Boxes. You receive a Supply Box for every 3 wins in Quick Fights, after being promoted to the next stage of the Championship or to the next League and by buying them in the in-game shop for gems. You will also receive a special Supply Box as a gift from Steevy from time to time (just because he loves you).
You can have up to 4 unlocked Boxes in your garage. Tap on a Box and then tap "Unlock" to start unlocking it.

The time it takes to unlock depends on the rarity of the box. You can skip the timer by tapping "Open Now". Note that each skip will cost you a certain amount of gems. The amount depends on the remaining time and the rarity of the box. You can also speed up the process by watching ads.

Tap on the unlocking box to watch an ad. Each ad will shorten the waiting time by 30 minutes.