The National Union of Teachers (NUT) has warned the Government to stop "playing with the educational future of this country" and scrap plans for "free" schools.
General secretary Christine Blower said the state-funded schools were "not wanted or needed" and parents had not been given enough say on the matter.
The Department for Education (DfE) said it was disappointed the NUT continued to "blindly oppose" the moves.
Free schools can be set up by charities, universities, businesses, educational groups, teachers and groups of parents, with the first 25 scheduled to open across 22 local authorities from September.
An NUT-commissioned YouGov survey of 1,021 parents in the approved locations found 31% were against setting one up in their area, while 26% were in favour and 29% were neither in favour nor against.
Ms Blower 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. The teaching profession and parents know this. It is time the Government stopped playing with the educational future of this country based on nothing more than the fact they can."
A DfE spokesman said: "It's disappointing to see the NUT continuing to blindly oppose free schools before one has even opened its doors just as they are blindly opposed to academies - schools which have proved incredibly popular with parents and pupils and have turned around under performance in deprived areas.
"As well as teachers and charities it is also parents themselves who are behind many of the free school proposals - parents who want something better for their children. And even where parents are not the lead proposers, each proposal has to show there is demand locally for the type of education they plan to offer.
"Of course parents want schools with playgrounds, libraries and kitchens and so do we. There is no reason why free schools wouldn't have these facilities but we want to be flexible in the buildings they can use. This is because there are excellent public buildings sat empty when new school places are needed. It's madness to spend tens of millions of pounds on new buildings in these circumstances."