Earlier this month, Park View Academy in Birmingham was one of five put in special measures.
Ofsted said it had not raised pupils' awareness of the risks of extremism, but vice-principal Lee Donaghy denied that when quizzed by MPs on Tuesday.
At a meeting on Sunday, many parents called on governors to resign.
Liam Byrne, Labour MP for Birmingham Hodge Hill, said there were about 200 parents at the meeting, and that they "overwhelmingly" voted in favour of governors stepping down.
He said they had called for fresh elections to let new governors have the chance to remedy problems.
"The message was that teaching is outstanding [at Park View] but there are problems with leadership, so let's change the management team," Mr Byrne said.
Park View vice-principal Mr Donaghy was one of several people to give evidence to the Home Office Select Committee on Tuesday.
Asked by its chair Keith Vaz if the position of the governing body was "untenable", he said it "would be a great shame, a great tragedy" if it were to go.
He also told the committee that he had heard differing accounts of the parents' meeting.
In a statement, governors at the Park View Education Trust said: "The governing body that helped drive the changes - that saw the school go from special measures to outstanding - is largely the same governing body that is here today."
The governors said they agreed that improvements needed to be made, but they also said that the inspection by Ofsted had been "prejudiced".
"In that climate it is unsurprising that parents might state that the governing body should resign, but as individuals we feel that we have been part of the solution, and are not the problem," they added.
A senior leader at one of the schools involved in the "Trojan Horse" inquiry said it would be a "great tragedy" if governors stepped down.