Under China's population controls, instituted more than 30 years ago, couples who have more than one child must pay a "social upbringing" fine, while in some cases mothers have been forced to undergo abortions.
Authorities in the eastern city of Wenzhou are investigating how the infant ended up beneath the vehicle, a Mayu county official surnamed Zhou told AFP.
"The family was agitated," he said. "After starting the car to bring the family to the office to discuss the matter, the official discovered the child had been crushed underneath the car."
The baby was rushed to hospital but could not be saved, the People's Daily reported on its website, adding that at the time of the accident the mother was sitting in the car and the father was getting in.
No details were available on how many children they had.
China says the one-child policy -- which exempts some rural families, ethnic minorities and couples who are both only children -- has prevented overpopulation and boosted economic development.
But some experts have called for the restriction to be phased out as the country's labour pool shrinks and the ranks of the elderly swell, while rights groups criticise what they call harsh enforcement methods.
There was widespread outrage last year after a woman who had been forced to abort seven months into her pregnancy was pictured with the bloody foetus.