If you have a level 6 part of a specific vehicle model, when it is upgraded to level 5, it can be put into the upgrade menu like other levels of parts (that is, the upgraded parts must be placed in order). These parts can be purchased from Donna, a "rare imported part", or they can be obtained by winning the race and disassembling the vehicle.
To view the level 6 parts you have, click the "(i)" information button on the right side of the garage, then click the item bar and select the "Level 6 upgraded parts" tab.