The parents of a teenager who died in custody have called for a public inquiry into the number of deaths of vulnerable youngsters in the UK's penal system.

Jake Hardy, 17, was one of two teenage deaths in prison last week. He was found hanging in his cell at a young offender institution in Wigan and died on Tuesday. Hours later, 15-year-old Alex Kelly was discovered unconscious at a juvenile prison in Kent and was pronounced dead the following day. Both had previously been identified as at risk of suicide or self-harm.

Gary and Elizabeth Hardy, from Chesterfield, Derbyshire, have issued a statement to the Observer demanding to know why their vulnerable child was seemingly able to kill himself in state care.

"We are shocked and devastated by the death of our son," they said. "Jake was a loving, caring person who was also vulnerable. There have been too many deaths of young people in prison. We support the calls for a public inquiry to understand why this is happening. We don't want other families to have to go through what we are going through."

Their criticism comes amid growing disquiet at the number of children handed short sentences who are being sent to prison because of cuts to traditional support services. Hardy was serving six months for affray and common assault at Hindley young offender institution, while Kelly, who was at Cookham Wood institution, received 10 months for burglary and theft from a vehicle. Both were taken to hospital, but could not be resuscitated.

Their deaths bring the number of child deaths in penal custody in England and Wales since 1990 to 33.

Deborah Coles, co-director of Inquest, a charity that advises the bereaved relatives of people who die in custody, said that the deaths of Hardy and Kelly raised fresh questions about sending vulnerable children to institutions unable to deal with their needs.

"This is a serious child protection issue and must be seen in that context," Coles said. "Investigations, inquests, academic research and prison inspections all point to a prison system that is ill-resourced and ill-equipped to deal with the complex needs of our most vulnerable children and young people."

The latest deaths coincide with newly disclosed evidence that corroborates growing concern about the mental wellbeing of young people in the prison system. A restricted report, commissioned by the Youth Justice Board, which has responsibility for young people in prison, suggests that the vast majority are unhappy in young offender institutions, the type of prison where Hardy and Kelly were both being held.

The report highlights a poll that found that only 28% of those held in young offender institutions and 32% in secure training centres agreed that it was the best place for them to be held, compared with 65% of those held in a secure children's home.

Of those in young offender institutions, 68% said they did not spend sufficient time, or any time at all, with their key worker, compared with 12% of those in a secure children's home.

Across all three types of institution, 27% of young offenders said that they were worried or scared about where they were held, while 22% reported that they had been bullied.

The findings will not be published until after the government outlines its strategy for the secure estate next month, increasing fears that the views of young people will be ignored or that they will be placed in unsuitable surroundings. Since 2009, the YJB has cut 36 secure children's home beds, reducing the total number of spaces to 183. On Tuesday it will brief councils on its plans to decommission a further 17 places.

Enver Solomon, chair of the standing committee for youth justice and director of policy at the Children's Society, said: "It is exceptional for two children to die in a matter of days and raises serious questions about what the prison service was doing to help those who are in its care.

"These children were clearly more of a risk to themselves than to anybody else, yet were locked up in prisons which are not child-focused and ill-equipped to look after those who are so vulnerable. It is time for the government to stop imprisoning children in institutions that fail them."

Coles added that the deaths underlined the need for a broad-ranging public inquiry to examine the policy of sending young people to prison.

"Successive governments have claimed 'lessons will be learned' and yet the deaths continue," she said. "Consideration of the wider social, structural and policy issues and the pressing need to learn from the systemic failings that have cost these children their lives needs to be addressed in a joined-up manner through a properly resourced public inquiry."

Campaigners have been demanding a public inquiry into the treatment of children within the juvenile justice system since the death of 16-year-old Joseph Scholes, a deeply disturbed youth who hanged himself at Stoke Heath YOI in Market Drayton, Shropshire, almost a decade ago.