A Bhutanese family who have lived and worked in Australia since 2012 are appealing to the immigration minister to give them permanent residency after it was refused because their deaf son would be a “cost” to the taxpayer.

The Wangchuk family live in Queanbeyan, in New South Wales, and are facing deportation this month unless the immigration minister, David Coleman, uses his discretion to grant them visas.

The family came to Australia in 2012 so the mother, Jangchu, could study in Melbourne, after which they moved to NSW.

In March the administrative appeals tribunal upheld a decision not to give the family permanent residency visas.

The tribunal member, Jennifer Cripps Watts, found the family did not pass the public interest requirement of the visa process, which stipulates the applicant be free of a disease or condition which would likely require healthcare or community services which would likely “result in a significant cost to the Australian community in the areas of healthcare and community services”.

“I know my son is deaf, but it’s not that he’s going to incur a huge amount of spending of taxpayers’ money,” Jangchu Wangchuk told Guardian Australia.

“He’s just deaf and uses a hearing aid and he never has any ongoing treatment or medication or anything. My husband and I are working so hard, and we are paying taxes. We have a very stable jobs and we love our jobs.

“He just goes for a yearly hearing test, that’s it. It’s not that someone has to take care of him, he’s quite independent, I don’t have to do anything for him.”

David Randall, a retired teacher from Melbourne, first came across the family when he was a teacher for the deaf at Kinley’s school, and has been advocating for them to stay in the country.

Randall, who started a petition to garner community support for the family, said when their bridging visas runs out this month the family won’t be allowed to work for the period Coleman is considering their application.

Kinley and his 17-year-old brother Tenzin are both in high school, and Jangchu works in childcare while their father Tshering recently began full-time work in aged care.

“They’re both trained nurses but their qualifications aren’t recognised here,” said Randall.

“They’re employed under one of the work-sponsored visa categories but part of that in terms of the residency component is that they all have to pass the health requirements.”

The tribunal hearing in March heard Kinley’s condition was likely permanent and would probably require both state and commonwealth disability services.

Watts said she had heard evidence that the family and its supporters believed the state costs of caring for Kinley would be “minimal” and that he had shown success at school since arriving in Australia.

“The applicant said that when they lived in Bhutan Kinley did not communicate effectively and now he does, by signing and in other ways as well,” she said.

However Watts said under the requirements of the act she had to uphold the decision to refuse their applications. She also decided against making a specific recommendation to refer the case to the minister for intervention.

Randall said the family had been contributing taxpayers while in Australia, and would be able to support Kinley.

“Kinley arrived essentially without education and is quite language disabled,” he said.

“No one who meets Kinley sees him as a burden. They see him as a contributing member of the population. He’s just that sort of young man. He’s cooperative, keen to learn, friendly, sharing and caring. They just make wonderful citizens.”

More than 28,000 people have signed a change.org petition calling for minister Coleman to intervene in the Wangchuk’s case and grant them visas.

“Everyone knows we’re going through such a difficult time but all this support, it makes it a little bit easier for us to bear,” said Jangchu Wangchuk.

“People are so supportive and I think they understand our story … Some organisations are even doing a rally tomorrow to help us. We don’t know any of these people but they are very good people.”

The midday rally, calling for the government not to deport the family, will be in front of Parliament House in Canberra.

A spokesman for the department said the health requirement was not condition-specific and was an objective assessment which affected all members of a family group even if only one person failed it.

“Ministerial intervention is not an extension of the visa process,” the spokesman said.

“A person is able to request intervention, however the minister cannot be compelled to exercise his powers and he is not required to explain his decisions on any case. What is or is not in the public interest is entirely a matter for the minister considering each case on its own merits.”