Police said officers opened fire Wednesday after trying unsuccessfully to subdue the 50-year-old man with capsicum foam at a cricket oval on the Mornington Peninsula, the Australian Broadcasting Corp. reported.
Paramedics called to the sports ground treated the 12-year-old boy but were unable to revive him, ABC said.
ABC said the father was flown to Melbourne's Alfred Hospital, where he was said to be in a life-threatening condition.
It was not immediately clear how the boy received his head injuries.
"The exact circumstances surrounding the incident are yet to be established," a police statement said.
Witnesses said the scene at the sports field was chaotic as children were attending cricket and soccer practice.