Two young sisters were orphaned when the family's car smashed into the back of a parked lorry, an inquest heard.
Simran, aged five, and 18-month-old Kiran, were seriously injured in the crash in Cambs which killed their parents Nishan and Perminder Singh.
The Cambridge inquest heard that none of the family, from Canterbury in Kent, had been wearing seatbelts.
The inquest was told Mr Singh swerved off the A14 near Swavesey into a lorry parked in a lay-by.
Mr Singh, 31, a village postmaster, and his wife, 28, who was sitting in the back of the silver Volvo with her daughters, suffered multiple injuries and were certified dead at the scene on 21 May this year.
Flung into footwell
But Kiran and Simran, who was flung into the front passenger footwell, survived and were treated at Addenbrooke's Hospital in Cambridge.
Coroner David Morris said it was not clear why Mr Singh had veered into the lay-by, but suggested he could have been tired, distracted or mistaken it for the service station.
The family are believed to have left home in Rough Common, near Canterbury, at about 2030 BST on 21 May, but it is not known exactly where they were heading.
Oliver Rumsey, from St Ives, Cambridgeshire, said he was riding his motorbike westbound along the A14 when he saw the Volvo on the inside lane at about 2220 BST.
He said he saw it brake for the speed camera, then accelerate, before swerving off the main carriageway into the lay-by.
He said: "It went nose-down into the back of a stationary lorry.
"I can only assume the driver must have fallen asleep or had a blow-out which is why they pulled into the lay-by."
Alan Hardie, the lorry driver from Birkenhead, who was not injured in the accident, said he had parked in the lay-by at Swavesey, went to bed at 2000 BST, but was woken with a start at 2230 BST.
Pc Clive Holgate said the Volvo had been travelling at above 65mph and was severely damaged.
Passing verdicts of accidental death for Mr and Mrs Singh, Coroner David Morris expressed his sympathy to the family.