Almost half of school districts in England will have more primary pupils than places within two years, the Local Government Association says.
The “unnecessary restrictions” on councils wanting to open schools could leave parents scrambling for places that do not exist by 2015, the head of the LGA told the BBC.
LGA chairman David Simmonds said that councils were facing “unprecedented pressures”, leaving schools to face a “desperate shortage” of places in the near future.
The warning comes as the government announces 93 new free schools opening across the country, creating an extra 43,000 spaces for primary and secondary school children.
But the LGA is calling for the Department for Education to work more closely with local councils, so planning for emerging demand for places can be better managed.
Its analysis of local authority data suggests about 1,000 of the 2,277 local school planning districts will be over capacity by 2015-16.
“Local councils have a legal duty to ensure there is a school place for every child in their area but they are being hampered by uncertainty and unnecessary restrictions,” Mr Simmonds said.
“This could result in parents scrambling for places that just don’t exist and threatens to seriously impact on our children’s education.
“The process of opening up much-needed schools is being impaired by a one-size-fits-all approach and in some cases by the presumption in favour of free schools and academies.”
Mr Simmonds added that free schools should only be allowed to open in areas in need of new places; five free primary schools appear to be launching in areas forecast to have surplus places several years from now.
Overall two-thirds of local authorities expect to be oversubscribed by the beginning of the 2016 academic year, the LGA told the BBC.
The greatest pressure is focused on about 99 districts, where 20 per cent more pupils are predicted than places will be available.
Peterborough, Redbridge, Ealing, Bristol, Lewisham, Slough, Maidenhead, Sutton and Barking and Dagenham will have to increase the number of school places by at least 20 per cent to ensure every child gets a place, the LGA said.
However, the association noted that the recent figures, submitted in 2012-2013, might not take into account recent steps to rectify the projected shortfall of places.
Education Secretary Michael Gove told the BBC that the coalition government had “taken swift action to repair the damage” caused by Labour.
“We have more than doubled funding for new school places and we are also setting up great new free schools, which are giving parents a choice of high quality school places in areas Labour neglected,” he said.
“Ed Miliband is too weak to apologise for the shortage of school places his government left behind and too weak to stand up to the unions and back free schools.”