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, the first 25 being scheduled to open across 22 local authorities from September.
An NUT-commissioned YouGov survey of 1,021 parents in the approved locations found 31 per cent were against setting one up in their area, while 26 per cent were in favour and 29 per cent 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."
The NUT accused the Government of lacking transparency after 76 per cent of parents said they were unaware a free school was planned in their area and 72 per cent said they had had no opportunity to have their say.
While 52 per cent of respondents said local councils were an appropriate body to run a school, 43 per cent said a group of teachers, 30 per cent a not-for-profit charity, 25 per cent a group of parents, 16 per cent a faith based group and 15 per cent a private company.
A total of 78 per cent said children in such schools should only be taught by qualified teachers, while 81 per cent believe the headteacher should hold a teaching qualification. And 72 per cent of parents thought free schools should follow the national curriculum.
The survey found 46 per cent of parents said there was no need for a new school in their area, while 28 per cent said there was some need and 19 per cent said there was a strong need.
Asked about essential facilities, 83 per cent said libraries, 86 per cent an outdoor play area, 78 per cent dining halls, and 69 per cent kitchens. The NUT said with free schools able to set up in shops, offices and factories, many will not have the space to provide such amenities.
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."