You can win races in a better car and use the "send parts to the garage" option to ship your reward to your garage. Parts sent to the garage can be retrieved and applied to other vehicles. However, this takes time and isn't a very effective way to accrue car parts. Instead..
If you've completed the story in this game then you undoubtedly have a lot of money. Take your car to a Tune-Up Shop and purchase the parts that you need. The parts available are based on the quality of your car you drive and usually hover 5 - 10 points above your current vehicle's level (approx 260 - 270 for your GT-R). Instead, purchase Catch-up packs with in-game currency. Theses packs can be purchased in tiers and max out at level 16 gear, around LV 300.
If you spend enough time at the Tune-Up Shop you can quickly upgrade a LV 250 car to LV 325+ without races via catch-up packs, shopping and trade-ins. Total cost is usually around 300,000 but that can be recuperated quickly. By this point you can chose how you want to fully spec-out your car: either by continued shopping or via races now that you car's competitive enough to participate.