A consultation has been launched by Millthorpe secondary school over its plans to combine with Knavesmire and Scarcroft primary schools to form an academy trust.
The school said opting out of local authority control would give the school more freedom.
Bob Scrase, who opposes academy status, said more discussion was needed.
Academies are independent, state-funded schools, which receive their funding directly from central government, rather than through a local authority.
Mr Scrase, who stood for Labour in the recent local elections, said more than 500 parents had sent postcards to the council calling for a ballot on the proposals.
"We think this is one way of raising the debate, said Mr Scrase.
"Getting more discussion going and getting a more widespread view of what parents think."
He added: "If we look at the risks and rewards, the risks of doing this [becoming an academy] seem to me to be much greater than the rewards that are being presented to us."
In its proposal document, Millthorpe School claimed opting out of local authority control "would allow us greater freedom to build the curriculum around the needs of our children."
The school also said it could not ignore that academy status might "provide opportunity for additional funding".
A consultation with parents will take place over the summer, after which the schools' governors will vote on the proposal.
If approved, the schools could become an academy by November.
Campaigners in York are calling for a vote on plans to turn three of the city's schools into academies.