Guy Ayscue smashed head-on into a car on US-1 last December.
He was killed, as were 18-year-old Linsay Lunsford and her nine-year-old sister Maggie.
Highway Patrol says Ayscue was trying to get away from Franklinton Police Officer Michael Dunlap, and that both reached speeds of up to 90-miles-per-hour, which violated Franklinton Police Department's chase policy.
The town of Franklinton, the town's police department and Guy Ayscue's estate are named as defendants in the case.
Clinton and Mary Ann Lunsford filed the claim seeking a jury trial and damages of more than $10,000.