A survey organised by the Foundation for the Study of Infant Deaths (FSID) revealed that only 37% of parents knew the room temperature should be kept between 16 and 20 degrees Celsius.
And when asked to guess how warm it was in their own home, just 41% of parents were able to estimate the temperature to within two degrees.
Researchers have shown that babies are at greater risk of cot death when they become too hot.
It may be that the internal thermoregulatory system in some babies is too immature to function properly and cool them down when the room temperature is high.
But the FSID's study of over 200 parents found that only one-third keep a thermometer in the room where their baby sleeps.
FSID director Joyce Epstein said: "Our message to parents is to look at and touch your baby to see if they are too warm, and keep an eye on the room temperature. Why take chances when creating a safer sleep environment for your baby?"
Worry about cold
Carol Bowman, whose daughter Nicole died of cot death 10 years ago, said many parents think that as long as they put their baby to sleep on their backs they will be safe.
"It's got to be rammed home to parents that the temperature is another thing they have to consider," she said.
In the UK's biggest-ever cot death study, carried out between 1993 and 1996, the mothers of babies who died were more likely to have worried about their baby getting too cold, while mothers of babies who survived were worried about the baby becoming too hot.
Around eight babies die every week in the UK because of cot death, most of them aged under six months.
Babies are more at risk if they are put to sleep on their stomachs, if their parents are smokers and if they become too hot.
The best way to tell if a baby is too hot is by feeling the tummy.
If it is hot or sweaty to the touch, then a layer of clothes or bedding should be removed.
Most excess heat is lost through the head, so the baby's head should not be covered by bedclothes.