The move, which is part of government plans to make PSHE compulsory in all maintained schools, is intended to provide all young people with information on issues such as body image, personal finances, alcohol and drugs, and sex and relationships.
It follows the recommendations of Sir Alasdair Macdonald, headteacher of a top performing inner-city multi-faith secondary school, who led an independent review on PSHE earlier this year.
In the past, parents had the right to withdraw their children from such lessons up to the age of 19.
But Balls said it is important that all young people receive at least one year of sex and relationships education (SRE) before their 16th birthday.
He said that research undertaken by the Department for Children, Schools and Families found that four out of five parents think all young people should receive SRE.
Balls explained: "A clear majority supported a reduction in the age to which a right of parental withdrawal should apply. Setting the age limit at 15 offers the best chance of building a strong consensus."
Sir Alasdair Macdonald said PSHE plays a vital role in preparing young people to deal with real life issues.
"I am pleased the government is taking forward my recommendations, including that schools should involve parents in developing their SRE policy. I support the government's decision to limit the parental right of withdrawal, given the importance of SRE for all young people," he said.
Simon Blake, national director of the sexual health charity, Brook, said statutory PSHE would make "a huge difference" to young people.
He said: "This announcement demonstrates that government has listened to the overwhelming majority of children, young people, parents, carers and teachers."
