Well, your parents are largely right and also quite wrong on two different things.
Boiling is fairly effective at killing most of the bacteria and all the viruses that can cause "food poisoning", but will not denature some of the toxins produced by some bacteria that can cause illness after consuming. Clostridium species spores are not inactivated by boiling, but are not waterborne and only found in anaerobic conditions such as in the gut of animals. If your dad had spilled gut contents over the the meat before cooking, I would be a little worried, otherwise, not so much.
Assuming that the meat and other ingredients were not contaminated with these bacteria to a significant level before cooking, boiled stocks should be fine on the counter for a overnight after cooking if they are left undisturbed and covered. Depending on the volume in which they were boiled and the ambient temperature, I would be surprised if the boiled meats were below 60 C (generally considered the temperature at which waterborne enteric pathogens are killed) after 2 hours, and as such would be too hot to place in the fridge.
However, they are wrong about the air exposure bit - if the meat/water was exposed (i.e. not in a covered container) to the air overnight, there is a good chance that there will now be some bacteria and yeasts happily growing in the stock produced by boiling meat. Most of these bacteria will not cause illness, but many can cause "food poisoning" and should be treated with caution.