The Born Free foundation say the deaths of a number of animals in recent years shows there were failings which the inspection system did not address.
It comes after the body representing zoos around the UK announced it is investigating animal deaths at the safari park near Stirling following complaints.
At the weekend, the Scottish Sun reported two lionesses died after being mauled by a male lion which was introduced into their pride in 2016.
One of the lionesses, Makalu, was pictured lying dead in a shed.
Deaths of a lemur and a nilgai antelope – both of which were attacked by other animals – are also being investigated, the newspaper reported.
The British and Irish Association of Zoos and Aquariums (Biaza) says it is investigating complaints about the park.
Dr Mark Jones, head of policy at Born Free, said: “Details of these shocking incidents, coming as they do just days after the killing of two brown bears at Whipsnade Zoo who escaped their enclosure and injured a boar, once again bring into serious question the ability of zoos in the UK to keep their animals safe.
“It seems the tragedies at Blair Drummond are indicative of systemic problems that have been ongoing for some years, but which the zoo inspection system has failed to address.”
Previous inspections had not raised concerns about animal escapes or problems with “compatible social groups”.
Dr Jones continued: “We urge the Scottish Government to launch a fully independent, transparent investigation, and to develop requirements and inspection processes for zoos that will ensure these kinds of tragedies can never be repeated.”
A Scottish Government spokesman said: “While it would be inappropriate to comment on an ongoing investigation, it’s important to note that we take animal welfare very seriously and are committed to ensuring the highest standards in Scotland.
“We are currently working with the UK and other devolved governments, and the Zoo Expert Committee, to review the ‘Standards of modern zoo practice’ document, which will be consulted on in due course.”