When you get an ultimate part for the first time, one part appears in your inventory. All subsequent copies will be added to that part's progress bar as 1/1 or 2/5, for example. When the bar is full, the "Upgrade" button becomes active.

Once you have collected the required number of part copies, you can upgrade that part with a special currency (purple banknote with a star). To upgrade a part, tap on it and then tap "Upgrade".