Around 63 per cent of parents surveyed said they have put money away for their children - and around half (51 per cent) of those saving started putting aside a pot of money at birth for their first baby.
Meanwhile, 33 per cent of parents said that grandparents help to top up their children’s nest eggs.
Mothers were slightly more likely than fathers to say they had started saving for their children, with 63 per cent of women saying they had done so - compared with 62 per cent of men.
While children’s saving accounts, Child Trust Funds and Junior Isas were the most popular products among those saving, one in 25 (4 per cent) savers said they were keeping cash at home.
Looking across Britain, parents in Scotland were found to be the most likely to put aside savings for their kids, with three-quarters (75 per cent) doing so - compared with 53 per cent in the South West of England, 56 per cent in the East Midlands and 61 per cent in London.
In Wales, 64 per cent of parents had put money away for their children.
Paying university fees and helping their children on the housing ladder were the most popular reasons for saving given by parents, with 21 per cent saving for each of these reasons.
A further 12 per cent of parents said they were saving to help their child buy their first car, and 6 per cent were saving towards school fees.
Anthony Browne, chief executive of the BBA, said: “Savers have been hard-pressed in recent years due to low interest rates.”
But he said that with recent speculation that interest rates could start rising gradually in 2016, now could be a good time for people to consider boosting their savings.
More than 500 parents with children aged 18 years old or under took part in the survey.