Sergio Rojas is from Mexico and his partner Linda Martinez is from Nicaragua.
The couple have a seven-year-old boy who was born outside of Canada.
Their three year-old son Jonathan was born in Yukon and is a Canadian citizen. He has a malformed skull which requires medical care.
The couple had applied for residency in Canada under compassionate and humanitarian grounds but were denied. They've been told they have to leave the country on March 4.
Pediatricians and therapists have told the couple their son won't receive proper medical care if they return to Central America.
'You have two choices'
Rojas says an immigration officer in Vancouver told the couple they could leave Jonathan in Canada.
"Ten days ago immigration called me and says 'you have to leave, you have two choices. We know he's sick, but you have two choices: You can leave him here, or you can take him."
"This is not a choice for us," said a tearful Rojas.
"We cannot leave members of our family. But they say leave it or take it. This is not a good situation for us."
Jonathan has been diagnosed with plagiocephaly and torticollis which is a type of muscle paralysis. He receives treatment at Vancouver's Sick Kids Hospital and gets daily therapy at the Child Development Centre in Whitehorse where he's learning to walk and talk.
"He's in a special program eight hours a day, he receives all the care here. And the pediatrician told me to don't move out," said Rojas.
The family has been in Canada for several years. Rojas is from Mexico but his family now all live in the United States.
Martinez has no family in Nicaragua.
Rojas says he can't understand why his three-year-old son is being penalized.
"Jonathan was born here. He needs you. He needs the government," said Rojas. "We don't ask for money, we don't ask for anything. Only for his health."
After the CBC aired the story on Friday reaction on social media was swift.
Yukon MP Ryan Leef met with the family twice during the weekend. He says the family had first sought assistance from his office in September.
Leef says he'll meet with Immigration Minister Chris Alexander on Monday and pledges to fight for the family's right to remain in Canada.
"Everything I've seen to this point, they've been compliant with all the requests that Canada has made, they've been diligent in their efforts to provide the information and follow the process. I'm identifying some gaps in the information that could be beneficial to their case. Our office is going to work exceptionally hard to help them work through something that they would have a challenge working through on their own."
Leef says it's a serious case with a very urgent deadline.
"The compassion and humanity that we can show this family, this Yukon family, this Canadian boy, warrants that. There's no question in my mind. If there's ever a time to stand up for a family or an individual, this one is very clear."
Sergio Rojas and his eldest son, Sergio Jr., leave for Vancouver today.
Martinez will follow on Wednesday with Jonathan.