A new poll, conducted on behalf of parenting website Mumsnet, found the majority (82%) of new mothers were unhappy about the practice.
The news comes after NHS officials were attacked for letting salespeople have access to new mothers just hours after they have given birth.
Some parents are approached by promotion company workers while they are still on postnatal wards, Glasgow-based GP Dr Margaret McCartney said.
In an editorial, published on bmj.com at the end of May, Dr McCartney wrote that Bounty, which gives parents free "baby bags", profits by selling parents' details to other companies. But she said the hours after birth were "hardly an optimal time" to obtain consent for giving out data.
The Mumsnet survey, of 1000 mothers who gave birth in the past year, found more than half of new mothers felt the Bounty rep invaded their privacy and three-fifths said they were not specifically told their details would be passed on.
The website has called on the Government to ban Bounty sales reps from marketing to new mothers on hospital wards.
"Our users believe it's time for the Government to clamp down on Bounty's harassment tactics to obtain data from new mothers," said Mumsnet chief executive Justine Roberts. A Bounty spokeswoman said: "We are saddened to hear of any individual situation that suggests we have fallen short of the high standards that mums demand of us and we demand of ourselves."
A Department of Health spokesman said: "It is up to local NHS hospitals to decide locally whether to allow Bounty representatives on to their maternity wards."