Barry and June Steenkamp appeared on the United Kingdom television programme Good Morning Britain and claimed news that the six-time Paralympic Games gold medallist is now eligible for parole after serving half of the sentence was a "horror story".

They revealed Pistorius has written them an "emotionally distressing" letter as part of South Africa’s "restorative justice" parole process.

Speaking on Good Morning Britain from their home in Port Elizabeth, June Steenkamp cast doubts over whether Pistorius was really sorry for the killing of their daughter.

"He's shown no remorse and he would only show remorse I think if it contributes to his getting out of jail," she told the programme.

"We haven't had the full story.

"I don't believe that it's the truth, he changed his mind three times under oath to a different story and we don't believe that's the truth of the story.

"That's what we want, the truth.

"This has been a horror story for us."

The 34-year-old Pistorius has been eligible for parole since July after he was convicted of murder for shooting Steenkamp multiple times through a toilet door at his home in Pretoria.

He claimed he had mistaken the 29-year-old model and law graduate for an intruder but was found guilty of manslaughter and sentenced to five years in prison.

The sentence was increased in 2015 to 13 years and five months after he was found guilty of murder on appeal.

Pistorius may now be released on parole after serving six years in a low-security jail in Pretoria.

As part of the parole process, he penned a letter to Mr and Mrs Steenkamp, which their lawyer described as "like ripping a Band-Aid off a wound".

"June and myself, we were dumbfounded," Barry Steenkamp told Good Morning Britain.

"I think the letter was sent to us to do with the parole that is coming up and yes, it's only June and I and our lawyer that has seen what has been written in it and we've kept quiet, and we don't want to push it out to anyone as to what was said.

"We might let people know at a later stage but not now, what was written by Oscar."

The Steenkamps are pushing for a personal face-to-face meeting with Pistorius.

"One day I would like to talk to Oscar, man to man," Barry Steenkamp said.

"I'd like to talk to him, June and myself we feel there are still a lot of things to come out of this story and we're hoping that Oscar will tell us the truth, and that's the reason why I'm the one who would like to speak to Oscar.

"I would like to have a chat to him one day and we'll see how it goes from there."

June Steenkamp made it clear they do not believe that Pistorius has spent enough time in prison yet.

"He must serve whatever his full time, in my eyes, he still must pay for what he's done," she said.

"And that's what we're expecting."

Good Morning Britain presenter Susanna Reid asked the parents if they could ever forgive their daughter's killer.

"Sorry is not enough in any case," June Steenkamp answered.

"It's not enough for losing her life, her future was ahead of her.

"Now she has no wedding, we have no grandchildren, and she never would have her wedding dress.

"She had nothing.

"He took everything away from her and us and that is something that is important to me, the things that she would never have had.

"I forgave him through God because I believe in God. 

"I'm a Christian, and I had to.

"God was asked to forgive him, but it doesn't mean that he hasn't got to pay for what he's done.

"He showed no remorse, and we don't believe the story.

"We don't believe the story that he gave.

"We don't know how it came to that.

"We want the truth."