Amanda Spielman told school leaders that it was “unacceptable”, adding that it would be a “huge step backwards” if schools became reluctant to teach children about the “diversity of modern Britain”.
Her warning came after parents have repeatedly held protests outside primary schools in Birmingham, which teach about the existence of same-sex parents.
Speaking at the National Association of Head Teachers (NAHT) annual conference in Telford, Ms Spielman said the reports of protests outside school gates and pupils being withdrawn from classes were “worrying”.
She highlighted that the actions could affect “community cohesion”.
At the same conference, school leaders warned of an “organised campaign” of protests against LGBT+ classes across the country.
Andrew Moffat, assistant head at Parkfield Community School, one the schools that has faced protests over LGBT+ lessons, said he had received a death threat from demonstrators.
Addressing hundreds of school leaders, Ms Spielman said: “It’s unacceptable to intimidate schools and to intimidate teachers who are trying to do what’s asked of them under the law.”
She added that she hopes dialogue will help to “remove misconception” and find solutions.
Her speech came after Sarah Hewitt-Clarkson, head at Anderton Park Primary School, which has faced daily protests over equality teachings, called on education secretary Damian Hinds to back schools.
Ms Hewitt-Clarkson, who has been subjected to chants outside the gates calling for her to resign and whose teachers have called the police after feeling harassed, asked the minister for support on Friday.
Addressing the head’s concerns at the conference, Mr Hinds called for the protests to stop and said children and teachers should not have to walk past demonstrations on their way to school.
He told the conference: “I back you to do your jobs, to make the right professional choices in the best interests of all your pupils and your teachers. And I expect you to be able to do this free from intimidation by others.”
Delegates at the NAHT conference voted in favour of a motion calling for schools to be supported when carrying out their duty under the Equality Act – which aims to protect people from discrimination.
New curriculum guidance, which has been updated for the first time in nearly two decades, on relationships and sex education expects secondary schools to teach students about LGBT+ content.
The Department for Education encourages primary schools to teach children about different families, including those with same-sex parents, but there is no specific requirement to do so.
During a debate on teaching about LGBT+ relationships on Saturday, Ms Hewitt-Clarkson called on the government to urgently work with headteachers, adding that “equality is always appropriate”.
School leaders warned that protests against LGBT+ classes have spread. Dave Woods, an NAHT national executive member, said headteachers in Ealing in London had received a number of emails, letters and requests for withdrawal from parents.
Mr Woods told the conference: “One head reported 20 parents surrounding him and demanding an on-the-spot meeting. Another one said a request for withdrawal had gone from two children to 80 in six weeks. This is happening across the country. It is an organised campaign.
“Relationships are for everybody. We don’t consult on maths, we don’t consult on English, we don’t consult on any subject, we don’t need to consult on relationships.”
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