The Scottish Parent Teacher Council (SPTC) said a new independent Education Tribunal Service would be the best way of handling families' concerns which had not been resolved by councils.
The call follows a Scottish Government consultation on changes to Section 70 of the 1980 Education (Scotland) Act.
Under Section 70, parents or pupils can complain directly to the Scottish Government about failures by their school or education authority. If the complaint is upheld, ministers can make an order through the Court of Session holding councils to account.
However, the Scottish Government has proposed scrapping the right of appeal to ministers, arguing that it takes too long. In future, officials want complaints to be handled by the Scottish Public Services Ombudsman.
A consultation response from the SPTC describes the present appeal system as "a slow, painful and labyrinthine process for families who have often spent years struggling to ensure their children receive an adequate education".
The organisation has called for complaints to instead be directed towards a new body - based on the existing Additional Support Needs Tribunal which deals with disputes concerning pupils with additional support needs.
The SPTC said: "We believe the one point of entry for parents who wish to pursue a complaint against a local authority should be a renamed Additional Support Needs Tribunal service, which has the understanding of educational issues and has a robust procedure in place to deal with such disputes.
"The tribunal, which we suggest be renamed the Education Tribunal Service, has expertise combined with clear and focused procedures which, we believe, give a much more robust and transparent route for parents."
The SPTC said that since the majority of Section 70 appeals already relate to additional support needs, such a move would not fundamentally change the remit of the tribunal.
It concludes: "We strongly recommend this alternative proposal to the Scottish Government as we believe it is sound in terms of equalities legislation, public service good practice and, ultimately, for parents and their children."
The SPTC intervention follows concern about the proposals from the Faculty of Advocates, who argued the right of appeal to ministers was a "vital safeguard" for families. A statement from the faculty published last month said the rights of parents and pupils could be "severely undermined" if Section 70 were to be repealed.
It added: "Section 70 provides a vital safeguard. It allows the ministers as an independent entity, appropriately advised by experts, to come in and examine what a local authority is doing and to support and enforce the rights of parents and pupils.
"It is a well-established safeguard, the removal of which has not been adequately argued for or justified."
A spokesman for the Scottish Government said: "We are in the process of analysing consultation responses and will issue a response later this year."