Strict new rules on term-time holidays - including fines - were introduced in 2013 to punish parents over absences.
But the LGA says the current system "does not always favour families".
Campaigners say a "blanket ban" remains unworkable, but schools minister Nick Gibb said the government was determined to raise pupils' attainment.
The fresh call from the LGA follows a case last week in which a father avoided prosecution for refusing to pay a fine for taking his child out of school for a holiday.
Source: Department for Education
A Freedom of Information request to councils by the Press Association found 86,010 fines had been issued by 98 councils in 2014-15 for pupil absence - either because of holiday or truancy.
The figure was up from 62,204 in 2013-14 and 32,512 in 2012-13.
Last week, the case against Jon Platt, 44, was thrown out at Isle of Wight Magistrates' Court after he argued the law required parents to ensure their children attended school "regularly", and did not put restrictions on taking them on holidays in term time.
He had refused to pay a £120 fine for taking his six-year-old daughter out of school to go to Disney World in Florida.
Craig Langman, who founded the organisation Parents Want A Say, which campaigns for the law to change, said 93% of all school absence was down to truancy.
"Only 7% is down to term-time holidays. So it's a massive sledgehammer to crack a nut," he said.
"At the end of the day, take the law back to what it was in 2013, when head teachers had the discretion of up to ten days on a case by case basis."
The LGA says families often struggle with the high cost of holidays out of term-time.
Councillor Roy Perry said family holidays can have "social and emotional benefits which are of lasting value and support to children".
"It should not be something for which they are unduly punished," he said.
However, schools minister Mr Gibb said it has "always been the case that you should not take time out of terms to go on holiday".
He said taking children out of class can disrupt teachers' planning, affect the education of other children and affect the child themselves.
"Our data shows that just a week off per year leading up to the GCSE courses can reduce the chances of that child getting good GCSEs by about a quarter," he said.
A "common sense approach" should be applied to parents in England taking children out of school for holidays, the Local Government Association says.