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Pretoria (CNN) -- Reeva Steenkamp's parents said Tuesday they are "satisfied" with the sentence handed down to their daughter's killer, as Oscar Pistorius started his first full day in a South African prison.

June and Barry Steenkamp told broadcaster ITV's "Good Morning Britain" show that they accepted the five-year prison term Pistorius was given Tuesday -- although they feel there's still a "big missing piece of the puzzle" concerning their daughter's final moments.

"He's got to pay for what he's done," said June Steenkamp.

"It's not that we want vengeance or anything or him to suffer with his disabilities, but at the same time we feel satisfied that he will realize now that you can't go around doing things like that."

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"We have gone along with the judge and her decision," her husband said. "Only Oscar knows whether that sentence is acceptable to him; I've got my feelings to the whole thing but we do accept what the judge handed down."

Pistorius, who won global renown as a medal-winning double amputee track star before his dramatic fall from grace, woke up Wednesday morning in Kgosi Mampuru II prison, formerly known as Pretoria Central.

He was convicted of culpable homicide, or negligent killing, but cleared of murder. His uncle, Arnold Pistorius, said the family accepted the sentence and that his nephew would "embrace this opportunity to pay back to society."

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'More to the story'

Asked if she believed the athlete's account that he shot Steenkamp through a locked toilet door under the mistaken belief that she was an intruder, June Steenkamp expressed skepticism.

"No -- there's more to the whole story than everybody knows, only Oscar knows," she said.

Her doubts are based in part on the evidence heard in court that Steenkamp was at times scared of Pistorius.

"She described exactly how she felt and that he wasn't treating her properly, he was always irritable with her and snapped quickly, so we know all of that so that's why we feel that it is not actually what happened," said June Steenkamp.

The couple were disappointed, "even shocked," that Pistorius was cleared of murder, she said, but feel that perhaps insufficient facts were put before the judge.

Barry Steenkamp told "Good Morning Britain" they had known "very little" about their daughter's relationship with Pistorius before she was killed on February 14 last year.

If an appeal is lodged over the sentence, they will have to accept it, hard though that may be, his wife said.

The key thing, she said, is that he will now realize in prison what he has done.

"I think he has lost everything but now he has to pay for what he has done to our daughter and no words can actually describe what that has done to us -- but we have to live with this now for the rest of our lives without her and it's so difficult to go through this and to have that pain in your heart and in your soul for her."

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Blankets, place to sleep

Ofentse Morwane, of Gauteng Correctional Services, told CNN that Pistorius was one of nine disabled offenders now held at the prison, notorious during the Apartheid era for its harsh treatment of political prisoners.

After leaving the Pretoria courtroom where his seven-month trial played out before the Steenkamp family and world's watching media, Pistorius was driven in an armored police van to the prison.

There, he will have undergone a health risk assessment within six hours of arrival, Morwane said. Within 21 days, a comprehensive assessment of his needs will be done.

"He was allocated blankets, a place to sleep," Morwane said. "The second day he will undergo an orientation and induction process which includes explaining to him the services available to him in the correctional center and ensuring he knows the dos and don'ts."

The prison has 7,217 inmates in total, just over 4,700 of whom have been sentenced, with the remainder in custody awaiting trial, Morwane added.

The defense argued during the sentencing hearing that Pistorius was too vulnerable to be sent to prison and should be sentenced to house arrest with community service.

But Judge Thokozile Masipa ruled that the prison system was equipped to cater for Pistorius' needs.

Orange jumpsuit

Prison commissioner Zedilon Monama said Tuesday that Pistorius was placed in a cell for one person in the hospital section of the prison. The cell has has a bed, toilet, blanket, sheets, prison-issued toothpaste and toilet paper, she said.

Pistorius was dressed in the standard-issue orange jumpsuit.

As is routine for all prisoners, he saw a nurse, psychologist and chaplain, Monama explained. Pistorius is a "B" group inmate, which means he's only allowed two no-contact visits on weekends, with a limit of 45 visits per year.

If he behaves, after six months, he could be allowed to join group "A" and get 60 visits a year, Monama said, and enjoy special perks like chocolate and Coca Cola.

CNN's Diana Magnay reported from Pretoria and Laura Smith-Spark wrote and reported from London. CNN's Khushbu Shah contributed to this report.