Nineteen people, mostly soldiers, were treated for light or moderate wounds in the incident in central Jerusalem.
Off duty soldiers shot the 19-year-old driver, in what Israeli police have said was an attack.
"My son was murdered, they killed him. He did not carry out a terrorist attack" said the driver's father.
"This was a car accident. The car stopped after hitting a wall. Why did they kill him?" Mahmoud Mughrabi said at his home in Israeli-occupied East Jerusalem.
Israeli police have said they were "100% sure" Qassem Mughrabi intended to carry out a deliberate attack, with one spokesmen saying a failed romance may have been the trigger.
Israeli police spokesman Shmuel Ben-Ruby said investigators had found Mr Mughrabi, who had no prior police record, "wanted to marry his cousin but when she refused he apparently decided to carry out the attack".
Relatives confirmed the family situation, but said Mr Mughrabi had told them he was going out with friends and would be back soon.
Then he took the keys of his brother's BMW car and drove off.
Mr Mughrabi said his son did not have a licence and could not drive properly.
He also wished a speedy recovered to the injured soldiers.
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