Teachers' union the NUT says a survey shows new free schools being opened in England are not wanted or needed.
A quarter of 1,000 parents said they would support a free school in their area, with 31% against and 43% unsure.
The schools - being set up by groups of parents, charities and faith bodies - will be funded directly by Westminster and be outside local authority control.
The government says the NUT is "blindly opposed" to the schools, which it argues will drive up standards.
The NUT is campaigning against free schools - and the conversion of existing schools to academy status.
Both types of school will be what the government calls independent state schools, with more freedom over the curriculum and teachers' pay and conditions.
Teachers employed by them will also not need to have formal teaching qualifications.
In September, the government gave provisional approval for 16 free schools to open at the start of the next academic year. Another nine have since also been given the go-ahead.
The NUT asked pollsters YouGov to survey just over 1,000 parents in 22 local authority areas of England where free schools are being planned.
When asked which groups should run schools, about half said local authorities, 43% said teachers, 30% said charities, 25% said parents and 15% private companies - people could specify as many options as they liked.
About half of those questioned said there was a need for a new school in their area - with the same amount saying there was not.
However, 31% said they were against or "tended to be against" a new free school. The largest proportion were neither for nor against such a school opening locally, or did not know their opinion on the issue.
Most parents questioned (72%) said they thought any new state-funded school should follow the national curriculum and that children should be taught by qualified teachers (78%).
Asked what impact they thought a free school would have on other schools in their local area, one in five said it would raise standards, while one in four said they did not know.
Christine Blower, NUT general secretary, said: "This survey clearly shows that parents are not clamouring to set up free schools, have no issue with schools being accountable to the community through democratically elected local authorities, and absolutely reject the premise of their children's education being handed over to private companies.
"Free schools are not wanted or needed. They are divisive and unaccountable.
"It is time the government stopped playing with the educational future of this country based on nothing more than the fact they can."
But Mark Lehane, a maths teacher planning to form a free school in Bedford, said he had the backing of parents who had opposed previous plans for a "ginormous" secondary school with up to 400 pupils per year group.
"What me and my teacher colleagues are proposing is a much smaller secondary school. There's an awful lot of community support," he told BBC Radio 4's Today programme.
"I'm really worried that the union leaderships are really opposed to their members trying to do something for their local community."
Asked about union concerns over the separation of free schools from councils' democratic structures, he added: "If unions wanted to set up free schools and build in these accountability processes involving the local council they have got the opportunity to do that."
And Rachel Wolf from the New Schools Network charity, which has received some government funding, said free schools need not have 100% support and would give parents more options.
"A lot of state schools are absolutely fantastic, but there are a significant minority that aren't," she said.
"Those are overwhelmingly in more deprived areas. What this survey shows is a very high number of parents think they don't have an option."
A spokeswoman for the Department for Education said: "It's disappointing to see the NUT continuing to blindly oppose free schools before one has even opened its doors.
"As well as teachers and charities, it's also parents themselves who are behind many of the free school proposals.
"And each proposal has to show there is demand locally for the type of education they plan to offer.
"Too often the poorest families are left with the worst schools while the rich can pay for good education via private schools or house prices. Free schools will give all parents, not just the rich, the option of a good local school with great teaching, strong discipline and small class sizes."