Vinnie was taken by ambulance to North Shore Hospital for initial treatment following the fall on Saturday afternoon, then transferred to Starship children's hospital where his arm was re-set under general anaesthetic.
Mrs Gee yelled at the aggressor, whom she estimates is aged 7, while attending to Vinnie and spoke to the woman with the boy.
"I said, 'If that's your child you need to know he's responsible for what just happened to my son'."
The woman did not appear concerned, did not speak to her and left the venue. Mrs Gee said she was disgusted by this.
"It was really cowardly the way she's just disappeared. I felt if that's how you behave, what hope is there that your child will grow up and be a decent human being."
In a Facebook post addressed to "The Mum of the Bully", Mrs Gee said the incident had converted an enjoyable birthday party for one of Vinnie's friends into a traumatic event
"The reason for this is down to your son's bully behaviour, and I deeply suspect, your parenting," Mrs Gee said in the post, which elicited responses of sympathy for Vinnie and condemnation for the woman's parenting skills.
Vinnie was discharged home from Starship in the early hours of Sunday and must wear a cast for six weeks, but was now "in good spirits," Mrs Gee said.
The venue's manager, Jarrod Armitage, said the incident was still being investigated. Two boys had been involved in shoving Vinnie. They had been told by a parent in his group, just prior to his fall, not to push other children.
The two left with the woman through a fire exit before staff could speak to them.
He did not know the woman's name, but CCTV footage was being checked for the number-plate of the vehicle in which she and the two boys departed.
He urged her to contact Inflatable World to discuss the incident, although the boys would be banned from returning. Five children have been banned from the venue for bullying, hitting another child or other bad behaviour in the three years it has been running.
Mrs Gee said more staff and other measures were needed to improve safety.
Mr Armitage said there were around 150 children, 37 adult spectators and six staff at the venue on Saturday.
When asked if that was enough staff, he said that although changes had been made to try to prevent a repeat of Saturday's incident, "There's certain things you can't control."
Worksafe said it was awaiting a response from Inflatable World after being notified of the incident by Mrs Gee.
"The file remains open, and no decision has yet been taken whether to undertake a site visit or launch a formal investigation."
