Councils must be given more powers to keep checks on schools and intervene when concerns are raised, according to the Local Government Association (LGA).
It said that in the wake of investigations into allegations of a "Trojan Horse" takeover plot in Birmingham, handing responsibility to councils to hold schools to account and offer support will "help to rebuild both parent and pupil confidence."
The association argued that local authorities should have the power to trigger Ofsted inspections, scrutinise budgets, challenge governors and provide help and support.
Cllr David Simmonds, chairman of the LGA's Children and Young People Board, said: "Whitehall acknowledges that it lacks the capacity and local knowledge to oversee the more than 3,500 academies and free schools in England and parents deserve a local organisation to act as a first port of call if they are not satisfied with the response from their child's school or its governing body.
"Mums and dads concerned about their children's education will always turn to their local council for help. But they are rightly becoming increasingly frustrated when their council is unable to intervene."
He added: "It is simply not acceptable that poor exam results or an Ofsted inspection should be the only trigger to identify how a school is performing. At present, there is a real risk of serious issues falling through the gaps and it will not be tolerated by parents or local government.
"We shouldn't have to wait until somebody blows the whistle to find out that something could have gone wrong. We need to be continually keeping a close eye on school performance."
The association said: "In the wake of investigations in Birmingham, the Local Government Association, which represents local authorities across England and Wales, firmly believes that empowered councils are the solution to holding local schools to account ensuring scrutiny, standards of conduct and support packages to help rebuild both pupil and parent confidence."
Under major education reforms, increasing numbers of schools have taken on academy status, meaning they are no longer under council control.
Local authorities do not have power to intervene in these schools, the LGA said, and at the same time they have increasingly less responsibility over the schools that are still under their control.
The LGA insisted that the only power councils have to intervene in schools without permission from Whitehall or a school being declared "inadequate" by Ofsted is to issue a warning notice.
It claimed that to exercise any further power, a local authority would need an Ofsted "inadequate" rating, for a school to have failed to comply with a warning notice or permission from the Education Secretary.
A Department for Education spokesman said: "Academies take power away from politicians and bureaucrats and give it to the heads and teachers who know their pupils best. Results are rising faster in sponsored academies than in council-run schools, and converter academies are more likely to improve their Ofsted rating.
"Academies are giving hope to children who have been failed by councils. Since 2010 we have taken 900 schools which were failing under council control and turned them into academies with the support of a strong sponsor. Thousands of other high-performing schools have leapt at the chance to take on academy freedoms."
He added: "We have consistently shown that we are tough on failure. We are strengthening the failure regime for academies through the new Regional Schools Commissioners and Head Teacher Boards. This will ensure swift action is taken in the small number of cases where academies are struggling."
Earlier this week, Ofsted published a damning verdict on the running of a number of state schools in Birmingham.
Five schools in the city were placed into special measures after a series of inspections in the wake of the "Trojan Horse" allegations.
The inspections, which were conducted following claims of a takeover plot by hardline Muslims, found that a "culture of fear and intimidation" had developed in some schools and, in several, governors exerted "inappropriate influence" over how they were being run.