Police confirmed the boy received medical treatment but he died at the scene.
An investigation was underway to determine the circumstances of the death.
Police said they have offered their deepest sympathy to the family during this difficult time.
A Christchurch City Council spokeswoman said the council didn't know if the child had been inside the stadium prior to the incident or if he was playing in a nearby playground.
Therefore she didn't want to comment and said it was a matter for the police to investigate.
SafeKids Aotearoa director Ann Weaver said the incident today was a tragedy.
"I do sympathise with the family for their loss and as well the driver," she said.
SafeKids Aotearoa has campaigned for drivers to be vigilant when reversing out of driveways if children are present. It was important to walk around your car before getting in to drive and the same message applied in carparks, she said.
"It's really important if you are in any situation, before you get in the car you walk around to see if there are any children," she said.
"People don't realise what they can't see when they are in a car.
"If you're reversing, you wouldn't see a 2-year-old behind you for up to 10m."
She said it was fortunate that incidents of children being hit by a car in a carpark were rare but it did happen from time-to-time.
"We often see this in different carparks whether it's a church or a supermarket.
"It's being conscious and realising cars aren't designed to see around it," she said.
Cowles Stadium in Christchurch's Aranui is a purpose-built, standalone indoor stadium, its website states.
The stadium is run by the Christchurch City Council with basketball, netball, volleyball and badminton courts for hire. It has an on-site carpark.