Seamus Fletcher, 2, cut his hand when he fell on a broken glass bottle at Fossil Grove in Victoria Park around a month ago.
The youngster was playing with his brother, his mum Aileen Monaghan and other members of the family when the accident happened.
They cleaned up the cut across his left hand and it has now healed.
But locals say the site, which has fossils dating back millions of years, has turned into a "magnet" for anti-social behaviour.
When the Evening Times visited, we found broken glass strewn across the area, and empty bottles of vodka, alcopops and cans.
Partick West coun-cillor Feargal Dalton, 41, who is Seamus' uncle, said: "My sister-in-law was upset and angry when it happ-ened. You don't expect your child to have his hand cut on glass when he's playing in a park.
"In the past couple of months it has turned into a drinking den for young people.
"Victoria Park has always attracted youths but now they have taken to drinking around Fossil Grove. There is broken glass there every Saturday and Sunday morning."
Mr Dalton said he had been liaising with community police to tackle the issue.
Councillor Kenny McLean, who is on the Fossil Grove board, said: "It has been a magnet for anti-social behaviour so we have made our represen-tations to the various authorities involved.
"People in the community are aware that it's happening and everyone is looking for action.
"Given it is a popular tourist attraction this behaviour does need to be nipped in the bud."
The incident came as a consultation into the future of Fossil Grove continues.
Inspector Bob Park, of Glasgow West End police, said: "We are aware of the issues and are doing regular bike and foot patrols in the area. We are trying to address the concerns."
A council spokes-man said: "We have highlighted this matter to our partner Community Safety Glasgow which is targeting suspected drinking dens."