More than 40% of parents admitted they had fallen asleep with their baby in a potentially dangerous way, such as on a sofa or in an armchair, which can increase the risk of SIDS by up to 50 times.
It comes as a new report from academics including from the University of Oxford found more needed to be done to raise awareness of safe sleeping practices.
It called for open conversations between parents and professionals, and looked in particular at how these messages can be conveyed to deprived communities.
Data suggests 42% of sudden infant deaths occur in deprived neighbourhoods, compared with 8% in the least deprived.
The Lullaby Trust said there needed to be much more information for parents, as it stressed the need to co-sleep as safely as possible.
If parents co-sleep, the charity says they should keep pillows and adult bedding away from the baby plus any other items that could cover their head or cause them to overheat.
Babies should sleep on their backs, other children or pets should not be brought into bed, and parents should ensure babies cannot get trapped, wedged or fall out of bed.
Babies should also not fall asleep with a parent on the sofa or in armchairs. Extra dangers are posed if parents drink alcohol, smoke or have a premature or low birthweight baby.
The Lullaby Trust poll found fewer than one in 10 (9%) parents who currently co-sleep with their baby had decided to do so before their baby was born.
More than half had fallen asleep in bed with their baby by accident, with 40% having done so more than once.
Jenny Ward, chief executive of the Lullaby Trust, said: “Co-sleeping needs to be discussed with all families.
It is really important that parents feel able to have open conversations about co-sleeping with health professionals so they get the right advice.
Most parents will co-sleep at some point, whether this is planned or unintentional. Simply telling parents not to co-sleep, or not discussing co-sleeping at all means that a lot of families could miss out on vital safer sleep advice that would help to reduce the risk of SIDS for their baby.”
Data from the National Child Mortality Database shows that, of all 6,503 infant and child deaths occurring between April 2019 and March 2021 in England, 30% occurred suddenly and unexpectedly and, of these, 64% had no immediately apparent cause.
Overall 126 babies were certified as SIDS deaths. Around half of babies who die from SIDS are co-sleeping with an adult or older sibling at the time of death, 60% of deaths occur when co-sleeping was unplanned and at least 92% are in hazardous circumstances, such as sleeping on a sofa or with a premature baby.