Parents believe the English education system provides too little choice and is too frequently run in the interests of politicians, not children, a think tank said today.
The poll comes with the release of a new report by the right-leaning think tank Policy Exchange which argues there is too much central government intervention in schools at the expense of the role of teachers.
The report, Helping Schools Succeed: a framework for English education, calls for radical changes in the English state school system.
It wants league tables to be scrapped and replaced with a school "report card" that would provide more information for parents.
The authors also suggest making all schools independent of local authority control and encouraging them to work with charities and not-for-profit organisations in the way academies already do to increase school choice.
Authors Chris Davies and Cheryl Lim examined successful school systems around the world, focusing on New Zealand, Canada, Hong Kong and Sweden.
They put forward ideas for replacing the national curriculum with a brief set of core subjects to allow schools more freedom, and back performance related pay for teachers.
The Policy Exchange says the findings of its survey show there is public support for the changes it puts forward.
The YouGov survey of parents and the general public found that two-thirds of parents believe that the national curriculum is manipulated to suit the aims of politicians and that three-quarters of parents want the curriculum to be set primarily by schools themselves.
More than 60 per cent of parents back performance-related pay for teachers.
Little over a third of parents regard the admissions system to secondary schools as fair or quite fair while just 14 per cent felt they had a lot of choice in terms of a school for their child.
It says just two per cent of parents decided which school to give preference to only or mainly on the basis of league tables and 42 per cent of parents did not consult them at all.
A total of 87 per cent of the population believe schools should be judged against a range of factors, not primarily exam results.
The head of Policy Exchange's Education Unit, Sam Freedman, said: "The national curriculum restricts the freedom of teachers and schools; it also holds back genuine diversity in the school system.
"The curriculum is increasingly cluttered with the latest fashionable subject discovered by ministers. It should be replaced with a brief core entitlement curriculum based on the Swedish model that will insist on minimum standards while offering flexibility.
"The lack of flexibility in teacher pay makes it difficult for schools in challenging areas to recruit and retain the best teachers.
"In conjunction with a new funding system that would prioritise these schools, local pay would allow the development of a labour market that benefited the poorest students. League tables don't work. They use information that can and is manipulated.
"They are far too complex for parents to use and they're terrible for teacher morale. We suggest a report card model that would be easy for parents to understand and would use far more sources of information than just exams."
* The YouGov survey of 2,161 adults was carried out online between February 27 and 29.