The inquiry, to be carried out by Tony Higgins, chief executive of the Cheltenham-based Universities and Colleges Admissions Service, is examining whether the governors followed the correct procedures in dismissing Peter Wilkes.
Mr Wilkes was dismissed after the school slipped from 147th to 205th in A-level league tables. Governors, who include General Sir John Waters, former deputy supreme allied commander Europe and Sir Michael Perry, chairman of Unilever, also criticised his managerial style and his relationships with senior staff.
But a meeting of parents voted by 620 to 7 in favour of Mr Wilkes' reinstatement and of the resignation of the school council (governors). They were also angry that they were initially given no explanation of the head's dismissal.
The case has highlighted the power of governors in private schools. All state schools are required to have elected parents on their governing bodies but independent schools are not. Cheltenham parents, who pay fees of around pounds 12,000 a year, say they should be told more about what is going on.
When parents first asked for an explanation for Mr Wilkes' dismissal from Air Commodore David Atherton, the council's secretary, they were told "matters between the council and its employees are confidential".
Later, governors sent a letter giving a more detailed explanation. They praised Mr Wilkes' gifts as a schoolmaster and his dedication to the school. They pointed out that the school's league table position was not the only reason for his dismissal, though they believed exam results were important.
Yesterday Mark Hicks Beach, chairman of the parents' committee, said the parents' demands were unchanged. "We are quite happy for Mr Higgins to investigate. We want to know the truth about the procedures behind Mr Wilkes' dismissal.
"We hope the inquiry will recommend that the constitution of Cheltenham College should be totally reorganised. There should be more local members and it should not be self-electing."
He said parents should have more communication with the council but he was not sure that necessarily meant having elected parent governors.
Two governors have resigned in protest against the decision to give Mr Wilkes two terms' notice, but the rest of the council cannot constitutionally be forced to resign.
Parents met five members of the council on Friday and a further meeting will take place on Sunday.
They fear that the change of head will mean a change of ethos at the school and that pupils who are unlikely to excel in exams will be thrown out.Reuse content