British National party activists have warned a primary school's headteacher and chair of governors that they will face demonstrations outside their homes if they do not drop plans to extend sex education lessons to children aged four.

A delegation from the far-right party picketed Grenoside Community primary school in Sheffield over proposals that would see children aged between four and six being taught about reproduction in mammals. Older children would be taught about the human body, including naming the sex organs, as well as receiving guidance on "good and bad touching".

BNP activists, who arrived as the school was preparing to close on Tuesday, demonstrated outside and handed in a letter warning the head: "We believe that your evil plans to introduce these children to sex at such a young age borders on paedophilia and that it is not acceptable."

The letter was copied to the school's chair of governors and Sheffield city council's executive director for children, young people and families. It warns: "We hope that you will see sense and we hope that no further action is needed, although we are ready to involve ourselves further on behalf of the silent majority if you insist on going further down your present road."

The BNP's leader, Nick Griffin, tweeted: "Small BNP team went to Grenoside Primary School, Sheffield today. Leaflets v plans for sex … lessons from age 4. Mental paedophiles behind sick plan must drop it or face demos at home too."

The school is consulting parents over changes that were originally scheduled for introduction in the spring or summer next year. The proposals will extend sex education to younger children.

The school's headteacher, Colin Fleetwood, said: "This is an attempt to teach a more systematic approach throughout the school. It's also more systematic about the science part of it."

He said that while a small number of parents did not want any sex education lessons in primary school, and others thought it was too early, the "vast majority" had been supportive of the changes.

For the youngest children, the lessons would involve being taught about baby animals, Fleetwood said. "It's mummy rabbits and daddy rabbits."

In response to the BNP demo, the school has sent a letter to parents saying the consultation will continue, and that the school's parents and governors will take a decision on the proposals.

Fleetwood said: "Obviously it's extremely unpleasant to have the BNP threatening to demonstrate, and receiving an ultimatum is not a pleasant thing. We're actually carrying out a very successful consultation with our parents, with a high level of engagement."

The Department for Education (DfE) recommends that all primary schools should have a sex and relationship education programme, but this guidance is not binding and it is up to schools to decide what to teach about reproduction and puberty beyond the science curriculum.

All children need to know about puberty before the onset of physical changes, the DfE says.

A DfE spokesman said: "Teachers have a right to do their jobs free from harassment, intimidation and abuse. The teaching of sex and relationships education in primary schools is a matter for schools and parents – it is not an issue for outside organisations to get involved in."