A Birmingham primary school that suspended lessons about LGBT rights following protests by parents should resume them, the former head of the schools watchdog has said.

Parents of pupils at Parkfield community school in Saltley have staged weekly protests over the lessons, which they claim promote gay and transgender lifestyles. Earlier this month, hundreds of mainly Muslim children, aged between four and 11, were withdrawn from the school for the day.

The school announced on Thursday that the No Outsiders lessons, which teach tolerance of diverse groups, including those of different races, genders and sexual orientation, would not be taught “until a resolution has been reached”.

Speaking to BBC Radio 4’s Today programme, Sir Michael Wilshaw – who was head of Ofsted until 2016 – said: “Conservative religious people [can] uphold their views … – sincerely held views – but also they’ve got to understand that they are living in this country, in a pluralistic society with liberal values, that strongly believes that people should be treated fairly and equally.”

Asked if he thought the lessons should be reinstated, he said: “Absolutely they should reinstate them.” Wilshaw said: “There have been problems in Birmingham for a very long time and it’s really important that there is strong political backing in Birmingham for this issue, from both the local authority, local MPs and also the Department of Education, to make absolutely clear that they will not tolerate this sort of behaviour.”

On Tuesday Ofsted cleared the school of any wrongdoing, praising Parkfield’s record on promoting “tolerance, acceptance and mutual respect” and confirming its Ofsted rating of “outstanding”.

A rally due to be held outside the school on Monday morning in support of “a LGBTQ inclusive education” was postponed, with the organiser, Sam Wise, saying he had decided to allow the discussions between the school and parents to continue without distraction. “The voices of hate and bigotry won’t go unchallenged. So demonstrations would be held to counter such activity if the need arises,” he said.

“However there cannot be any pandering to demands to halt lessons much longer, or remove No Outsiders completely. Equality is for all and LGBTQ-inclusive education is a matter of life and death. We cannot have another generation of LGBTQ youth feeling that they do not belong or do not matter.”

He said on Twitter that he had heard homophobic groups who didn’t have children at the school had been exploiting the situation for their own gain.

The Excelsior Multi Academy Trust, which runs Parkfield, announced that the lessons would be suspended last week following talks with parents and education chiefs. In a statement, the school said: “Nothing is more important than ensuring our children’s education continues uninterrupted.

“Both parents and the trust held constructive discussions with the regional schools commissioner, and, as a result of these discussions, we are eager to continue to work together with parents over the coming days and weeks to find a solution that will support the children in our school to continue their education in a harmonious environment.

“Until a resolution has been reached, No Outsiders lessons will not be taught at Parkfield and we hope that children will not be removed from school to take part in protests.”

In January, the school’s assistant headteacher Andrew Moffat said he was threatened and targeted via a leaflet campaign after the school piloted the No Outsiders programme. A petition signed by 400 predominantly Muslim parents called for the lessons to be dropped from the curriculum.

Moffat, who was awarded an MBE for his work in equality education, was shortlisted for a world’s best teacher award given out by the Varkey Foundation, but missed out to Brother Peter Tabichi, a member of the Franciscan religious order.