The actor was killed by his own car at his home in Los Angeles in June.
It struck him after rolling backwards down the steep drive, pinning him against a brick postbox pillar and a security fence.
Court records show Victor and Irina Yelchin filed to become administrators of their son's estate on Friday.
The filing states Yelchin left behind at least $641,000 (£486,000) in personal property and had $731,000 (£554,000) in equity in his home.
Yelchin was best known for playing Pavel Chekov in the new Star Trek films.
TMZ reports that Yelchin's parents plan to file a wrongful death lawsuit against Fiat Chrysler, which made the Jeep that killed their son.
They are expected to announce their intentions to file the suit at a press conference on Tuesday.
In April, the company  recalled 1.1m vehicles worldwide over the risk they could roll away after drivers exit.
The vehicle that killed Yelchin, who was 27 years old, was one of those subject to a recall because the gear shifters confused drivers, causing the vehicles to roll away unexpectedly.
Fiat Chrysler offered their sympathies to Yelchin's family in a statement, but said it could not comment on their plans for a lawsuit.
The company added it was speeding up its recall of the vehicle.
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Anton Yelchin did not write a will before he died and left behind a $1.4m (£1.06m) estate, according to court documents filed by his parents.