The van driver started his vehicle, thinking it was in neutral, while standing beside it, but it was "in gear and jumped back", he told Fairfax.
"At that very moment, the child has moved behind the van ... and he's unfortunately been pinned between the van and the low fence."
The driver of the van is believed to be the mother's personal trainer and she was standing metres away when the incident occurred, Fairfax reported.
Cowles Stadium in Christchurch's Aranui is a purpose-built, standalone indoor stadium.
The stadium is run by the Christchurch City Council with basketball, netball, volleyball and badminton courts for hire. It has an on-site carpark.
Police were investigating the incident but it was too early to say if charges would be laid against the van driver.