The research has also found that two in five parents believe every child should have an opportunity to go to private school, irrespective of family background and income.
The findings form part of a survey on social mobility commissioned by the Sutton Trust, which will form a manifesto – or series of recommendations – ahead of the General Election next year.
The report states that admissions need to be fairer, particularly in areas where schools are over-subscribed.
It found that England’s best performing schools are “significantly more socially selective” with lower numbers of pupils eligible for free school meals (FSM).
The survey found that more than a quarter of the 1,100 parents interviewed supported a ballot or random system of school admissions, with a further fifth believing that half of school places should be allocated in this way, with the remaining places allocated on proximity.
More than four out of 10 parents believed that all children should have a chance to go to a private school, regardless of family background, and at government or tax-payers’ expense. However, almost three in 10 were against this proposal.
The manifesto suggests one solution might be a means-tested voucher for poorer families to spend on extra tuition, cultural activities and books for children. Schools could also use new powers to give priority to pupils receiving the Pupil Premium.
Lee Elliot Major, the Trust’s director of policy and development, said that all political parties recognised the need to improve social mobility, and “fairer admissions and fairer access must be at the heart of any programme to improve social mobility”.
The Sutton Trust has also called for schemes, paid for by the state, which would open up leading independent schools to pupils, based on their academic ability rather than their ability to pay.
Nansi Ellis, assistant general secretary at the Association of Teachers and Lecturers (ATL), said: “While allocating places by ballot may seem fairer, it will make admissions to oversubscribed schools even more complex and is likely to lead to parents who are unsuccessful feeling their child is attending a ‘second best’ school.
“We believe that ballots and banding would be more successful in closing the attainment gap if they are carried out across a local authority area rather than for individual schools.
“We would urge closer investigation of both forms of allocation on a local area basis, and consideration of how best to ensure that all schools admit a broad range of children across a range of educational and financial situations.”