Call for private schooling for all
Two in five parents think every child should have the chance to go to private school, regardless of their family's income, it has been suggested.
Many other mums and dads are in favour of at least some state school places being allocated randomly, or by ballot.
The findings are part of a survey commissioned by the Sutton Trust, for its new "mobility manifesto", which sets out a series of recommendations ahead of next year's General Election on how to make sure all children have the same opportunities, regardless of background.
It says there needs to be fairer admissions to comprehensive, grammar and fee-paying schools when pupils are age 11, including through more use of ballots and banding, especially in urban areas that are over-subscribed.
Research has shown that England's top performing comprehensives and academies are "significantly more socially selective", with fewer pupils than average on free school meals - a key measure of poverty, the manifesto says.
This indicates that poorer youngsters are missing out.
A new survey conducted for the charity found that more than a quarter of the more than 1,100 mums and dads questioned supported all school places being allocated by ballot, or randomly, with a further 19% agreeing that half of places should be allocated in this way and the remaining places handed out based on distance from home to school.
Under the current system, parents apply for the state schools they want their child to attend, but may not get their first choice if the school is over-subscribed. State primaries and secondaries have to clearly set out the measures they will use to allocate places if they are over-subscribed, such as distance.
The poll also found that two in five (41%) of parents think that all children should have the opportunity to go to private school, regardless of family income and at the government or taxpayer's expense.
Nearly three in 10 (29%) were against this proposal.
The Trust's manifesto suggests that there needs to be better information about schools, and means-tested vouchers should be made available to poorer families to spend on extra tuition, books and cultural activities for youngsters.
All schools should use the powers recently handed to them under the new admissions code to give priority places to students who are entitled to the pupil premium - extra cash handed to schools to help disadvantaged children.
It goes on to call for the barriers between state and private schools to be broken down.
The Trust calls for support, including state funding, for a scheme to open up leading fee-paying day schools with pupils admitted based on their academic abilities rather than their family's ability to pay.
Participating schools would get the same funding per pupil as local state schools, the Trust said, but charges fees on a means-tested basis, with the poorest paying nothing and middle-income families given reduced rates.
Dr Lee Elliot Major, Sutton Trust director of policy and development, said: "There is a clear recognition in all the main political parties that we need to do a lot more to improve social mobility in Britain. Our ten point manifesto gives them the ideas that will turn the rhetoric into radical change for the better.
"Fairer admissions and fairer access must be at the heart of any programme to improve social mobility. Our poll shows a public appetite for change in oversubscribed comprehensives and academies. We need changes too to ensure fair access to grammar schools, independent schools and elite universities."
:: The new YouGov survey questioned 1,169 British parents of five to 18-year-olds between August 26-29.
Nansi Ellis, assistant general secretary at the Association of Teachers and Lecturers (ATL), said: "ATL believes that fairness in school admissions is absolutely vital, and is becoming increasingly difficult as the range of school types increases. 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.
"ATL believes 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."
A Department for Education spokesman said: "We strongly agree that all children, regardless of background, should have the same chance to get on in life. This is the principle at the heart of our plan for education, and we have already done a huge amount to close the attainment gap between advantaged and disadvantaged children.
"Policies such as the pupil premium - extra funding for schools worth £2.5 billion a year - and changes we have made to the Schools Admissions Code are helping disadvantaged children get the support, education and skills they need to get on in life and succeed.
"We want all children to be able to go to a good local school and, as a result of our plan and the hard work and dedication of thousands of teachers who have put it into action, more children have the chance to go to a good or outstanding school today than ever before".