The brothers' parents, Anthony Clark (63) and his wife Christine (61) said it was terrible to watch their grown sons deteriorate before their eyes and described feeling "powerless" to help them.
Mr Clark drew parallels between the disease and the 2008 fantasy drama, The Curious Case Of Benjamin Button, based on the F. Scott Fitzgerald novel, in which an old man, played by Brad Pitt, gradually gets younger.
"It's a devastating disease. Both of them are very childlike now," he said.
"Matthew went out the other day and bought himself a train set and a Mr Potato Head.
"We will be out walking and things which might interest a toddler interest them, the other day we were walking home when Michael saw a balloon and pointed it out to us.
"He also has these awful episodes where he screams and shouts and says, 'I don't know what I'm doing' because the disease effects his memory, he can't remember much of who he was.
"In the mornings we have a routine where we have to get a board game out and let him play with that so that he comes out of his 'state' as we call it.
"It is like an adult having a toddler's tantrum. It's obviously worse for him but it is terrible for us too. There's nothing we can do to help and we feel absolutely powerless."
Mr Clark, who is retired, said the men's lives began to unravel when their symptoms started to become more pronounced. They initially stopped replying to texts or returning phone calls but when Matthew had lost his job, he was so confused he sat in a room with just a candle at night and no food or electricity for two weeks because he didn't know how to claim benefits.
"He was as helpless as a child," his father said.
Michael, who had been discharged from the RAF the previous year after damaging his knee, was eventually evicted from his flat because he stopped looking after himself and doing basic household chores.
He slept in a park for three weeks before finding a soup kitchen which sent him to the Salvation Army. Staff there realised something was wrong and sent him to a doctor, who carried out an MRI scan and diagnosed the disease.
Mr Clark said: "When they asked him if he had any siblings he said he had a brother and they ran tests on Matthew and discovered he had the same thing.
"At first we were told doctors thought no one else had this condition and then we found out there are 100 adults in the UK, but no one had heard of it occurring in two siblings.
Mr and Mrs Clark, from Lincoln, retired to Benidorm in 2007 but returned to the UK in January to care for their sons.
Leukodystrophy refers to a group of neurological disorders caused by damage to the myelin sheath, the protective covering that surrounds nerve fibres in the brain.
When the nerve endings are damaged, the brain stops sending out correct messages and the body is unable to perform basic functions such as walking, talking and swallowing.
The genetic disorder can be passed down from a parent but can also mutate spontaneously.
The Myelin Project funds research into the disease in an attempt to find treatments and, ultimately, a cure, but the illness is so rare that little is known about it.
Lynda Carthy, chief executive officer of Myelin, said it was often misdiagnosed as the symptoms can resemble those of a stroke, Multiple Sclerosis or Parkinson's.
"There is an estimated one in three billion chance of two people who carry the gene deficiency meeting and becoming partners. There is no cure and it will eventually end in death as the brain shuts down."
They were diagnosed with terminal leukodystrophy last year.
Michael (42) served as a gunner with the RAF and once had a place on the parade display team. Matthew (39) married in 1990 and has a 19-year-old daughter who is expecting a baby. But his relationship crumbled when he began to display early symptoms of the disease.