The parachute component, from my understanding, is different than the manual cut off which simply cuts the fuel off. I believe that the parachute deployment is triggered by a change in manifold pressure. So in this case it doesn't apply.
I also thought, at one time, they talked about having the chutes automatically deploy if the driver hasn't done so a certain distance past the finish line. Not sure if that was implemented or not.
Edited...found it. The link to the NHRA website is no longer active but here's the summary:
The 2010 season brought a new safety device to Top Fuel classes. Should the driver be rendered unable to perform the normal shutdown sequence at the conclusion of a run, a pair of redundant transmitters, placed 400 feet (120 m) and 600 feet (180 m) past the finish line, will signal an on-board receiver to automatically shut off ignition power and fuel to the engine and deploy the parachutes. The transmitters are designed and placed so as to avoid inadvertent triggering of the automated shutoffs. These transmitters and the receivers that are placed on all cars were designed by NHRA's Track Safety Committee and constructed by Electrimotion, and are a direct result of Kalitta's death.
So I'm not sure where the posts above were going but the parachute "issue" didn't apply to Dixon's scenario.