Asylum seekers cannot be deported until their claims are processed.
The father has since withdrawn his refugee application because of deteriorating mental health in prison, says his immigration adviser Tuariki Delamere.
Delamere has been calling for their release from prison, saying they posed no threat to New Zealand's security.
The Herald has not been able to reach the affected family for comment.
The fear of Covid-19 and infectious disease on its own will not normally bring a person within the definition of a refugee or protected person, says an INZ spokesperson.
"They wouldn't qualify as refugees," said immigration lawyer Simon Laurent, who has been representing refugees since the mid-1990s.
To qualify, an asylum-seeker needs to demonstrate a well-founded fear of being persecuted for one of the five reasons cited in the 1951 Refugee Convention: race, religion, nationality, membership of a particular social group or political opinion.
"It will be interesting to see how the Refugee Status Unit processes these cases," said Raj Pardeep Singh of Legal Associates, whose services focus on ethnic communities.
"It's a new pandemic and of course people are dying, but it's treatable," he said, "If you look at Indian data the recovery rate is very high."
INZ says two of the 14 claims have since been withdrawn, meaning there are currently 12 individual claims being assessed by INZ's Refugee Status Unit.
Initial decisions can be expected early next year, but the Immigration and Protection Tribunal may make decisions relating to Covid-19 before then, the spokesperson said.
Laurent says there are policy reasons why authorities would want to "nip it in the bud".
"If you gave protection status to people because of Covid then you're opening the floodgates to millions of claims for people in hundreds of countries."
