The number of first-time buyers relying on family loans from the “bank of mum and dad” to fund their deposits is exacerbating inequality and impeding social mobility, a government-backed study has warned.

The Social Mobility Commission found the percentage of first-time buyers turning to family for financial help had increased from 20% in 2010 to a historic high of 34%. The report also found just one in three (31%) 25- to 29-year-olds owned a home compared with 63% of the same age group in 1990.

“Home ownership helps unlock high levels of social mobility but it is in freefall among young families,” said the former Labour MP Alan Milburn, the chair of the commission.

“Owning a home is becoming a distant dream for millions of young people on low incomes who do not have the luxury of relying on the bank of mum and dad to give them a foot up on the housing ladder. The way the housing market is operating is exacerbating inequality and impeding social mobility.”

The report, which used data from the University of Cambridge and Anglia Ruskin University, examined a series of surveys, including the English Housing Survey – which interviewed around 13,300 households in 2013-14 – and the Labour Force Survey, which looked at homeowners by age between 1995 and 2015.

It found one in 10 first-time buyers used inherited wealth and that 12%, whether it was a first property or not, were using a “gift or loan”.

First-time buyers who receive cash or at least a loan from their parents can buy 2.6 years earlier that those who do not, and this figure rises to 4.6 years in London.

The study also found 30% of households with dependent children held assets that could be used towards a home deposit, but only around 10% of households without any formal education qualifications over two back-to-back generations felt they could help out their kids.

The report predicts that if the economy weakens, the proportion of first-time buyers being propped up by family will remain at about one third until 2025, but it will boom to 40% by 2029.

Theresa May’s government earlier this year strayed from David Cameron’s “home-owning democracy” approach to housing and announced tougher regulation on landlords.

The proportion of people living in private rented homes has doubled since 2000.

“It is welcome that the government recognises the growing problem people face in getting on the housing ladder,” said Milburn, who was a Treasury and health minister in Tony Blair’s governments.

“A major national effort is needed to expand opportunities for home ownership and will require more radical action on housing supply.”

The commission’s State of the Nation 2016 report encouraged the government to build 3m homes over the next decade and build on the green belt.

The report’s lead author, Dr Paul Sanderson, said only “better-off” young people with parents who had accumulated housing wealth were likely to consider home ownership unless there were “radical changes to the housing market”.

“It is further embedding social immobility into the housing market,” said the Anglia Ruskin senior lecturer and University of Cambridge fellow.

“Obviously it’s down to affordability, increasing housing prices and incomes staying static compared to inflation.”

Sanderson said an increase in building social housing, regulation of planning developments and a focus on the help-to-buy scheme could help turn the tide.

John Healey, the Labour party’s shadow secretary of state for housing and a former government minister, however, criticised flaws in the scheme.

“This is further evidence of an inequality of wealth and opportunity for home ownership in this country and it’s part of seven years of failure on housing under Conservative ministers,” Healey told the Guardian.

“The government has got to do more to help young first-time buyers who don’t have the bank of mum and dad behind them.

“They could start by making sure help to buy is better targeted. It’s indefensible that help to buy is helping almost 25,000 people who are not first-time buyers.

“Younger people on smaller incomes are increasingly locked out.”