Transforming Lives for Good (TLG) was speaking after David Cameron announced the plans at the Conservative Party conference in Manchester.
Currently, a parent(s) or guardian is fined £60 if their child misses school without what the government calls "good reason". This fine increases to £120 if the parent/guardian fails to pay it within three weeks.
If this doubled fine remains unpaid after 28 days, the government plans to take it directly from the parent/guardian's child benefit.
David Cameron has said the plans will ensure all children get a good education and achieve their full potential, however teaching unions have said the fines will make the difficult situations of parents whose children are skipping school even worse.
He said: "We all know the evidence that if you don't attend school regularly you get a less good education, you get worse results, as a result your job prospects are much much worse and as a result your life chances - the opportunities you have to make the most of your talents - are severely reduced.
"That is why school attendance is so important."
"These penalties ... are only imposed where someone has been fairly persistently truant and they can't resolve this issue with the school.
"What we are saying today is where those fines aren't paid, they should be taken out of a parent's child benefit to make sure the signal is absolutely clear that it's your duty as a parent to get your children to school every day.
"That is a very strong positive message to send out and I think on behalf of taxpayers who fund this child benefit, it is quite right to say to people: yes, child benefit is your right but with that right come some responsibilities."
However Mike Royal, the National Director of TLG, told Premier's News Hour he was concerned: "By penalising those parents through the child benefit system it's trying to solve a social problem with the wrong tool.
"That would have the biggest impact on the child themselves and not just the parents.
"The first responsibility naturally is the parent's responsibility [to ensure a child goes to school]. There is also a responsibility on schools to ensure that if a child is disengaging, that the right kind of services is put around them.
"We have specific concerns around the idea that you penalise families and therefore children through the child benefit system."
Listen to Premier's Hannah Tooley speaking to Mike Royal on the News Hour here: