Gillian Penlington, a researcher at the centre-right think tank, recommends that early-years centres actively encourage parents to get involved.
And she adds that although affluent parents are more likely to volunteer, it is those from deprived backgrounds who get the most out of being involved.
She said: "We were doing research into the benefits to children of pre-school education, but ... looking at children in deprived communities it became apparent that some parents were getting real returns from helping."
Studies show 280,000 parents help in early-years centres, but Ms Penlington said this could be doubled.
Benefits to parents from volunteering include higher take-up of education and finding work, and less anxiety and depression.
The study, sponsored by the Pre-school Learning Alliance, comes as the Department for Education and Skills seeks tenders to develop and run a training scheme for Neighbourhood Nursery managers so theycan support parents who lack basic skills.
The Government is setting up 900 Neighbourhood Nurseries in disadvantaged areas over the next three years, which will provide 45,000 new childcare places.
Ms Penlington said: "We welcome this - but what about areas that Neighbourhood Nurseries don't cover? Centres in those areas should adopt a similar role reaching out to parents and their children."
"The Parental Stake in Pre-school Education", Social Market Foundation, price pound;5