LOS ANGELES, Aug 2 (Reuters) - The parents of the late "Star Trek" movie actor Anton Yelchin sued Fiat Chrysler Automobiles NV for negligence and product liability on Tuesday over the rollaway crash of a Jeep Grand Cherokee that killed their son.

Yelchin, 27, was killed on June 19 when his 2015 Grand Cherokee, which was under a recall issued by the automaker two months earlier, rolled backward in steep driveway of his Los Angeles home and crushed him against a brick wall and fence.

The lawsuit, filed in Los Angeles County Superior Court, is believed to be the first wrongful death complaint linked to the Fiat Chrysler recall in April of more than 1.1 million cars and sport utility vehicles because of vehicles that can roll away after drivers exit them.

The recall applies to 811,000 vehicles in the United States. It has been tied to hundreds of mishaps including 68 injuries, 266 crashes and 308 reports of property damage. Yelchin's death marked the first known fatality linked to the reason for the recall.

While not acknowledging a mechanical flaw in its transmission gear selectors, the company has said rollaways have occurred because drivers mistakenly believe they had placed the vehicles in park before getting out.

The automaker said in a statement it had not been served with a lawsuit and declined comment on the legal complaint.

But the company said it has urged its "customers to follow the instructions in their owners manuals and the information cards sent with their recall notices," including recommendations that drivers always set their parking brakes before exiting their vehicles. (Reporting by Piya Sinha-Roy in Los Angeles; additional reporting by David Shepardson in Traverse City, Michigan; Writing by Steve Gorman; Editing by Jeffrey Benkoe and Grant McCool)