Competition for primary places is particularly acute in some parts of the country.
While there were 548,006 applications for places this year, demand for primary places is still rising and this year went up to 641,572, a 0.8% increase on last year's figures.
Local authorities would be able to deal with the extra demand for primary places by building and expanding existing good schools, she added.
David Cameron demanded an end to the bitter coalition row over free schools, as Labour seized on Lib Dem claims that education secretary Michael Gove had raided the budget for primary places for an extra £400m to fund his "pet project".
In London, where demand for primary places is highest, one in six families learned on Monday that they had failed to get a place for their child at their preferred primary from September.
The union criticises the Education Secretary Michael Gove for spending on free schools rather than the pressing need for primary places.
This shows a picture of soaring demand for primary places in some areas, within a projected national increase in the number of primary-age pupils of 454,800.
As parents prepare for Friday's deadline in applying for a primary place for children due to start school in September, research suggests many will face disappointment when it comes to national offer day next April.
Parents and children will be able to visit the completed building before the deadline to apply for a new primary place for September 2020.
As well as lack of funding, barriers to the creation of primary places for all included conflict, poverty, discrimination, poor governance, corruption and rising population.
