To obtain nitrous for the drag race you must be on a day ending in a 6 (26, 86, 236, etc). You must first lose a drag race at least one day before the day ending in a "6", or up to 3 days in advance (ends in "3").

Once you have lost, and it is a day ending in a "6", go to the used parts shop and the part will be all the way to the right under the "???". The part costs $1980 and you can only use it once (by pressing Y button) in a drag race.