Two children who did not eat the meat are unaffected.
Doctors are understood to be puzzled at the dire impact of the suspected poisoning.
The hospital spokesperson said the "three patients are responding to botulism anti-toxin and are recovering in hospital".
A priest visited the hospital on Sunday to pray for the afflicted family and friends from the Hamilton Marthoma congregation have been making daily bedside visits.
Subi Baba, left, Shibu Kochummen and Shibu's mother Alekutty Daniel have been in Waikato Hospital for a week with suspected poisoning after eating wild boar meat. Shibu called an ambulance but he too fainted halfway through the call.
Family friend Joji Varghese told Stuff that a toxicology report is expected out November 17 that could give some answers about what the family is suffering from. "We have no evidence to believe there is any public health issue".
The region's Medical Officer of Health, Dr Richard Vipond, said it was investigating potential sources for the illness, including wild pork meat.
The family, who moved from Kerala to New Zealand five years ago, are currently settled in Putaruru on New Zealand's North Island and are now in a critical condition at a hospital in Waikato.
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"These were extremely active, full of life people and all of the sudden, nothing", he said, breaking down in tears.
The children were not fed the contaminated meat, as they were already in bed.
Medical specialists treating the three reportedly indicated it could take up to two months for the poison to clear their bodies and that they could face long-term damage including paralysis or tremors. They began to suffer about half an hour after they started eating.
The couple's two children - a 7-year-old and 1-year-old - are being looked after by the church group.
Their grandmother is on a visitor's visa from India and does not have medical insurance, so she will be billed full medical costs for treatment that doctors say could last up to six months.
The Indian High Commission and Ministry for Vulnerable Children have been approached to help the family. "We are doing our best to not let this incident affect them, but not sure for how long though", he said.
"If the dose is very high, the symptoms could be seen more rapidly-the incubation period could be the day of consumption", Pomerleau told the New Zealand Herald.
He said it wasn't the first time the family had eaten wild boar.