You get a "Burnout" when applying maximum acceleration from either a dead stop or when rolling in the opposite direction, so you can get this from a parked position by tromping the gas (either left or right trigger).
As soon as the burnout stops, you can get another by using the opposite trigger in the same way. So, when rolling forward, pull the left trigger all the way back and when rolling backwards pull the right trigger all the way back.
Note that the cool down period for burnout is fairly long, so you will not get one on each attempt unless you stop and wait a second, so you may prefer to do it that way.
This is a bit easier with overpowered or drift happy vehicles such as. the 1974 Toyota Celica tune named "230+MPH C-Class" by "Don Joewon Song".