A head teacher at a primary school giving lessons on LGBT equality has received threatening emails and phone calls.
Police are investigating messages sent to Sarah Hewitt-Clarkson at Anderton Park Primary School in Birmingham.
There have been seven weeks of protests outside the site from which “hundreds” of pupils were kept away on Monday.
Birmingham MP Jess Phillips has called for an exclusion zone at the school to limit where people can demonstrate.
Ms Hewitt-Clarkson has branded the protests aggressive.
The city council is looking into Ms Phillips’ request, with the authority’s leader saying some outside the school are “peddling hatred”.
The complaints at Anderton Park, mainly from Muslim protesters, focus on lessons for which pupils have been given books featuring transgender children and gay families.
The protests’ leader says that amounts to “social engineering”.
Similar teaching has been opposed in letters sent predominantly by conservative Muslims to schools across England, BBC Newsnight reported last week.
Ms Hewitt-Clarkson said of the protests: “There’s a whole variety of emotions: embarrassment for lots of our community and our parents who think this is just awful what’s happening; frustration that it’s going on so long; frustration that great British laws like ‘you can protest peacefully’ actually are causing us a problem.
“It’s interesting what a normal person on the street would think peaceful means and what actually is peaceful outside here.”
She described the scene in the Sparkhill area of the city as “very loud, it’s very aggressive, it’s tiresome”.
Ms Hewitt-Clarkson said she was “meeting lots of parents”, with a series of 12 meetings set up between now and the end of June.
She also denied a claim from some parents that she is Islamophobic, saying she believed in “equality for everybody”.
In England, relationships education will be compulsory for all primary pupils from September 2020.
Shakeel Afsar is the leader of the Anderton Park protests, although he has no children at the school.
He said the school had pulled “the shutters down” on parental engagement and was promoting LGBT lifestyles to children.
He said 600 pupils were kept from school on Monday “to make it crystal clear we will not have our children indoctrinated or participating in any social engineering programmes which undermine our family values by promoting child sexualisation”.
Anderton Park said more than half of the 700-strong student body had attended school. The council has been contacted to confirm attendance figures.
Overnight, counter-protesters adorned the site with heart-shaped messages featuring the words “love is the answer”.
West Midlands Police, which is investigating the threats against Ms Hewitt-Clarkson, said officers were also looking into “disorder” outside the school in which eggs were thrown at the counter-protesters.
The force said it was investigating three reports of assault and two of criminal damage.
Outside the school earlier, Mr Afsar was involved in a stand-up disagreement with Ms Phillips, Labour MP for Birmingham Yardley.
She said protesters could not “pick and choose” which equality they could and could not have.
Saying the worst thing about the protests was damage “to the reputation of a peaceful” community, she called for an exclusion area “to protect the 700 children in this school”.
Ian Ward, leader of Birmingham City Council, said he had asked authority officers to see whether they could use a Public Spaces Protection Order (PSPO) to counter the protests.
He said: “If a PSPO is not appropriate, then we will look at alternative options, because the children and staff at Anderton Park have a right to attend school without this daily disruption.
“It’s one thing for parents to ask questions about elements of a school curriculum, it’s quite another for others to pounce on the situation as an excuse to peddle hatred and misinformation.”
A council spokesperson said PSPO proposals would normally go out to public consultation and, based on response, a decision made by the authority and “police leads”.