There are 4 separate achievements related to "Winning a race with handling parts for "maximum" (off-road, drift , road and race) traction equipped" The key to earning these achievements doesn't have much to do with any specific part setup but how each part affects the particular vehicles handling "graph". I've provided 2 images of examples of a MAX build that would yield an achievement(s) and a build that would NOT yield an achievement.
NOT a max build:
https://imgur.com/XJ8auSsmax build in Road/Race:
https://imgur.com/rAXWQh8Pay attention to where the dot on that graph is located in order to setup your car for any of the 4 particular achievements. The ONLY parts that affect this handling graph are SUSPENSION & TIRES under the chassis category and DIFFERENTIAL under the drivetrain category. MOST exotic/sports cars in the game can achieve "max" handling in all categories with the exception of OFF-ROAD. So to put things simple just adjust the 3 parts I listed until the car reaches max in the desired category. For your off-road achievement I'd suggest grabbing a truck or SUV of some kind due to many cars being not able to achieve max off-road even with the best part rarities & yes rarity DOES matter.
Lastly, go and WIN a race with your new "max" vehicle in your desired category. I can also confirm that drift trials count as a race.