Parents want the ruling body of the General Teaching Council for Scotland reduced from 50 to 15 members, with a guaranteed place at the table for themselves.
Education directors, on the other hand, want an end to the current regulations that give teachers an inbuilt majority.
These were among the responses to the Scottish Government's consultation about the future shape of the GTCS when it becomes independent later this year.
The GTCS opposes a drastic reduction in membership, mainly because this would reduce the number of people available to sit on its various committees and reduce its effectiveness.
It is also resistant to diluting teacher numbers, given that teachers' annual fees pay for it.
Earlier this year, GTCS chief executive Tony Finn warned against using the General Medical Council as a role model.
The GMC has 24 members drawn equally from the medical profession and the public; the GTCS has 50 on its council, the majority registered teachers.
According to the Scottish Parent Teacher Council, the current GTCS council of 50 makes it a "very cumbersome body in governance terms" and difficult for individuals to make their voices heard.
It suggests a smaller council of 15 - a figure that was also advocated by the Association of Headteachers and Deputes in Scotland. The council could then be supported by a larger pool of qualified teachers who would provide the expertise to maintain the committees, according to the SPTC.
It has also called for the GTCS to be "more obviously independent, not only from government but also from the teacher unions".
It said: "To outsiders, the GTCS seems like an extension of the teacher unions, often walking in step with their views and policies."
It nevertheless recommended there be eight teachers and seven non-teachers on the council.
Secondary heads agreed it was "essential" the teacher majority be retained. School Leaders Scotland wanted no reduction in the size of the council
"GTCS is currently experiencing difficulties in servicing key committees," it said. "Our view is that a smaller council would compound these difficulties."
A council of 39 members was considered workable by the Association of Directors of Education in Scotland, but it saw no need for teachers to occupy the bulk of seats.
"There is no reason why elected teachers and other members shouldn't be of the same number, and this would emphasise partnership and collegiality," it said. "To insist on a majority of teachers hints at decisions by voting rather than consensus."
Parents, pupils or the business community must be better represented, it argued.
The consultation also asked about the GTCS's responsibilities - for instance, whether it should expand its continuing professional development role.
More than 100 responses were received from individuals and organisations, including teaching unions, local authorities, universities and colleges.
The Government is expected to announce its plans in the autumn.