To obtain new spare parts, you need to open Supply Boxes. You receive a Supply Box for each 3 wins in Quick Fights, after being promoted to the next stage of the Championship or to the next League and by buying it in the in-game shop for gems. You also receive a special Supply Box as a gift from Steevy from time to time (just 'cause 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 for unlocking depends on the rarity of the Box. You can skip the timer by pressing "Open Now". Note, that each skip costs you a certain amount of Gems. The amount depends on the time left and rarity of the Box. You can also speed up the process by watching ads. Tap on the unlocking box to watch an ad. Each advertisement cuts off 30 mins of waiting.