He has been issued an interim visa while his case is being considered.
"As Liam is not a New Zealand citizen he needs to meet the requirements to be granted residence," said Immigration area manager Michael Carley.
"INZ owes it to the New Zealand taxpayer to ensure that visa applicants have an acceptable standard of health to minimise costs and demands on New Zealand's health services."
Mr Carley said Liam's residency application was lodged on March 4 and had been referred to an independent medical assessor.
"Liam has not been disadvantaged by this as he has been kept on a valid visa to enable him to stay in the country," he said. "Some applications can take longer than others to process because of their complexity."
Mr Mahon has lodged a complaint against the agency, saying their decision to grant residency to just one of his two sons is "discriminatory".
"His brother was granted his visa in March, and three months later we still have no idea about what will happen to Liam," Mr Mahon said. "As they are both our children, we find this very discriminatory towards someone with special needs."
Williams syndrome affects one in 10,000 people and is present at birth, according to the Williams Syndrome Association.
People with the syndrome usually suffer medical problems, including cardiovascular disease, developmental delays and learning disabilities, and children can struggle with things like spatial relations, numbers and abstract reasoning and require early interventions, such as speech and occupational therapy.
Mrs Mahon said Immigration NZ's indecision had put the family under a lot of stress and Liam had fallen behind on his medical treatments.
"He's overdue for his heart check-up ... he also needs speech therapy, physiotherapy, occupational therapy and he needs to see the doctors," she said.
She said they could not afford to pay for the treatment privately. It would cost about $1000 per visit to see the cardiologist.
Mrs Mahon, who moved to New Zealand in 2006, said she returned to Malaysia to deliver Liam because she wanted to be close to family.
They then moved to Germany for two and a half years. Nolan was born there.
"It is unfair ... The fact that both of us are entitled to live and work here must surely mean that our dependent children should be able to live here with us without all this red tape," she said. "We really have no idea what we'd do if Liam doesn't get a visa to stay ... We don't have a plan B."
Fight to stay
• Liam Mahon, 3, has Williams syndrome, a genetic condition affecting about 1 in 10,000 people who require ongoing medical care
• Immigration NZ grants residency to his 13-month-old brother, but places Liam's case under consideration
• Immigration New Zealand says it owes it to taxpayers to ensure visa applicants have acceptable standard of health
• Liam's father, a New Zealand0 citizen, says Immigration NZ's action discriminates against people with special needs
