The number affected since protests were first staged at Parkfield Community School in Birmingham has "spread", according to The Sunday Times.
The newspaper spoke to leaders from a number of schools in the city and Manchester about strong opposition families have expressed about guidance recently brought in by the Government.
It makes teaching on relationships compulsory in primary schools and lessons on sex and relationships compulsory in secondary schools.
The curriculum, which teaches about LGBT lifestyles and same-sex relationships, has raised questions over whether children should be taught about certain subjects contrary to the wishes of their parents.
Parkfield has since dropped its Outsiders programme - an in-house programme incorporating books including one about a boy who dresses as a boy - following opposition.
In the face of fierce criticism from some parents, the school has called upon the Government for support.
It sent a letter to the education secretary Damian Hinds which said: "We are not allowed to mention LGBT... Bullying has increased and homophobic incidents have risen.
The Department for Education warned against protests frightening children and being "hijacked by vested interests."
A spokesperson told the Sunday Times: "There is no reason why teaching children about the different types of loving, healthy relationships that exist cannot be done in a way that respects everyone's views."
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