The attack comes as Taliban insurgents are pressing ahead with a nationwide offensive despite the onset of winter
No group has so far claimed responsibility for the blast targeting Fakuri Behishti, an MP from central Bamiyan province, who was on his way to parliament.
"The bomb was placed under a bridge and it detonated when the MP's vehicle was passing through," interior ministry spokesman Najib Danish told AFP.
"Three people including the MP were wounded."
But parliament speaker Abdul Raouf Ibrahimi told lawmakers that a relative of the MP had also been killed.
"I heard a big explosion that shook nearby shops. I saw rescuers pulling out several wounded people from the destroyed vehicle," said Abdul Manan, a a bakery shop owner near the scene of the attack.
The attack comes as Taliban insurgents are pressing ahead with a nationwide offensive despite the onset of winter, when fighting usually ebbs.
The militants have recently stepped up attacks on Afghan lawmakers.
Last Wednesday at least eight people were killed when Taliban suicide bombers stormed the residence of an Afghan lawmaker in Kabul, an attack that lasted nearly 10 hours.
Helmand MP Mir Wali survived the assault with injuries, but two of his grandsons and bodyguards were among those killed.