The left-wing party is putting forward a bill to ensure the content of the curriculum is 'factual and objective'
Fianna Fáil has been described as the "dinosaurs of the Dáil".
The claim came from Solidarity's Mick Barry following a debate of his party's Objective Sex Education Bill.
The legislation would ensure all schoolchildren get age appropriate sex education, including information about contraception, consent and LGBTQ issues.
The party wants to remove religious ethos from the discussion - and says the content of the curriculum should be 'factual and objective'.
Speaking after the Dáil debate, Deputy Barry argued: “Fianna Fáil were exposed as the dinosaurs of the Dail this evening with their opposition to a Bill to provide for objective sex education. Even the government in their speeches said that they would not oppose the Bill."
He claimed that the main opposition party opposes 'any interference' with religious ethos of schools.
The Solidarity TD added: "That obstacle [of religious ethos] must be removed with legislation, which is what our Bill does. Is Fianna Fáil going to block a new generation of young people having access to factual sex education?”
His colleague Ruth Coppinger, meanwhile, called for supporters of the bill to mobilise and persuade their TDs to vote in favour of it.
She said: "We can win by pressurising government TDs to vote Yes instead of abstaining and by pressurising Fianna Fail TDs to vote yes in opposition to the position put forward by their spokersperson tonight."
During the debate, Fianna Fáil's education spokesperson Thomas Byrne said his party supported many of the suggestions in the proposed legislation - but does not want the curriculum put into law.
He told other deputies: "This goes against all precedent. We think it is bad precedent to set.
"We think that children and young adults are best served by the National Council for Curriculum and Assessment - which is effectively teachers, educationalists and professionals who will seek expert views from outside groups... such as you have yourself, and I acknowledge that... and that they then will come up with the detailed curriculum."
Earlier this month, Education Minister Richard Bruton announced a major review of relationships and sexuality education (RSE) in schools.
Officials are being asked to consider a number of areas - including issues around consent, developments in contraception, LGBTQ matters, and safe use of the internet.