Yes of course it is the parents' responsibility to keep their children safe. As it is in the home. But to be blunt, if you don't take care in the home, it's only your child who will cop it.
The point about the toddler groups specifically, is that there are other adults wandering about who are not taking care.
The only poster on here who says that her child was scalded at a Mum and Toddler group clearly stated that it was someone else's mother who was careless. That woman was probably careful enough with her own children though.
What would the response have been to an AIBU along the lines of "A woman spilt hot coffee all over my DD at a toddler group and then said it was my fault for not watching her properly." ?
I personally agree that Mum and Toddler groups need to serve hot drinks. It's part of what they do and what they are.
But they need to take certain precautions, and a lot of them do not. I listed some of the things they can do to make things safer in an earlier post.
My main issue is the lack of compassion to the OP, who is clearly still terrified over the experience she relates.
When I approached the leaders of a Mum and Toddler group about my fears years ago they were so nice to me. Not much changed but they did at least try to accommodate my suggestions. Most of all, no one called me an idiot, or suggested I need to grow up. (They didn't even look at me as if I was mad, as did the Mum who's hot coffee was about to spill all over her child's head before I alerted her.)
A little kindness goes a long way, especially to someone who is already suffering.