Body
There are eighteen parts that can be unlocked on the body:
Front Bumper - Drift for at least 8 seconds
Grille - Locate and defeat your first Roaming Racer
Front Fenders - Complete 3 Jumps with at least 1 Star
Headlights - Locate the parts for, and complete, your first Derelict build
Splitter - Complete 3 Speed Runs with at least 1 Star
Front Canards - Locate and destroy 5 Billboards
Hood - Complete 3 Jumps with at least 2 Stars
Wing Mirrors - Complete 3 Speed Traps with at least 2 Stars
Roof Lights - Unlock at a Performance Level of 300
Side Skirts - Achieve 3 continuous seconds of Air Time
Spoiler - Drive at 175 MPH for 3 continuous seconds
Trunk Lid - Obtain Rep Level: 3
Diffuser - Complete 3 Drift Zones with at least 1 Star
Exhaust - Reach a speed of 195 MPH
Tail Lights - Drive for a total of 155 miles
Rear Bumper - Complete 3 Speed Traps with at least 1 Star
Rear Fender - Purchase a second garage
Wheels
There are four parts that can be unlocked on the wheels:
Tyre Bombing - Reach a 2x multiplier in Free Roam
Calipers - Complete 3 Drift Zones with at least 2 Stars
Brake Discs - Win your first Side Bet
Rims - Trade in 5 Speed Cards
Super Builds
I am not sure if these two are required for this achievement, but they are listed here in case they are (I had them when I unlocked the achievement). I used the Chevy Bel-Air as my Drag Build, allowing me to get not only the following two, but the Roof Lights listed above.
Super Build - Once a Derelict Build has a Performance Level of 300, you have the option of purchasing it's Super Build.
Wheelie Bar - Purchasing the Super Build for a Derelict in the Drag category will unlock this option