The National Association of Head Teachers (NAHT) annual conference in Telford was told national demonstrations were being held by opponents of children being taught about same-sex relationships.
Conference unanimously passed a motion calling on the national executive to work with appropriate bodies to develop and lobby for more robust policy and support for schools.
Dave Woods, a member of the NAHT national executive, said head teachers in Ealing, west London, had been sent a number of emails and requests for withdrawal from parents.
He said: “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.”
Earlier on Saturday Ofsted chief Amanda Spielman said it was “unacceptable” for teachers and schools to be intimidated.
She also said it would be a “huge step backwards” if schools became reluctant to teach about diversity in Britain.
Ms Spielman said: “The Equality Act is designed to enforce a number of different rights. And of course, there are places where these different rights can bump into each other.
“One clear tension exists in places where equality between the sexes comes second to religious belief and cultural preferences.
“Another tension arises between religious belief and relationships education in the context of LGBT issues. And what we’re talking about here is not sex education.
“It’s just a simple understanding that, just as families worship differently, they also love and marry differently.”
The chief inspector of schools added: “But as a result, we’re seeing protests at school gates and children being withdrawn from schools.”
Conference heard the demonstrations were worrying on a number of fronts including their effect on community cohesion and the impact on schools and children.
Ms Spielman said: “Clearly, it’s unacceptable to intimidate schools and to intimidate teachers who are trying to do what’s asked of them under the law.”
Three new subjects – relationships education from primary school, relationships and sex education at secondary school, and health education for all ages – will form part of the school curriculum in England from 2020.
Primary school children will not learn about sexual relationships, but will learn about having respect for all kinds of people in an age-appropriate way.
Where a maintained primary school chooses to teach aspects of sex education, the school must set this out in its policy and consult parents on what is to be covered.
Addressing the NAHT conference, Education Secretary Damian Hinds called on protesters to stop, and instead engage in dialogue.
He said: “No child should have to walk past a protest to go to school, and no teacher should have to walk past a protest to go to school.
“It is right that there is dialogue, we want schools to consult with parents. Ultimately there is no veto over what is taught in schools. That is head teachers’ domain.
“I call on people to not be protesting – talking by all means, but not protesting – because it is not good for those children, and it is not good for your fellow professionals.”