Okay, as for the 15 month old - No Way. They are too young to be left alone. Choking, climbing the stairs and falling. etc. Yes, there was a group of kids, but a parent cannot expect even a 6 year old to watch over a 15 month. -- let alone to always play nicely. That is putting adult responsibility on a child. Just DON'T DO IT.
Yes you can leave your 3 year old by themselves in a child proof room within earshot. And things can go wrong, but that is not a reason to shelter them. Too much sheltering will leave them afraid to make decisions on their own, scared of everything, and unable to interact with their peers.
As I said before, bad things can happen. My three year old actually pulled a dresser over by opening each drawer from the bottom up. That is right, the tiny kid dresser tipped over and she was under it. The drawers kept the dresser from crushing her. Many people secure dressers to the floor or wall, but I can bet that most do not. Now, she is my third child - the other two are much older and this is the first time this happened. From now on, I secure all furniture to the walls. - This shows that we live and we learn. Same as kids - They have to be able LIVE in order to LEARN. I still let my three year old go into her room alone, now we not only have the bookshelf and TV secured, but all furniture.
On another note - I remember my first Pediatrician saying, "And some came back home safely and the others, well they didn't." This was after I made a comment about how I have a hard time allowing my kids to go outside by themselve, but my parents let me and my siblings go outside all day by ourselves. So here is the thing, kids need freedom and they need supervision. Simply put, you cannot expect a child to always make the right decision. So don't leave them at the park by themselves. Go to the park with them, but sit on a nearby bench and don't get involved with every altercation. Let them handle it, and if it starts to get violent, then step in. But they have to learn to socially interact. AND they CAN do this with you near them, if you are a parent that tells them to try and deal with things on their own first. Plus, if you want, then take them to a recreational center for free play. There are adults there who supervise and many times you can leave your kids there. Plus, for the most part, the adults in the recreational centers let the kids work things out on their own. They only step in when they absolutely must.
If you want a specific age - most States do not have a specific age as to when a child can be left home alone, but the few States that do say 12 years old if the child is mature. If the child usually follows rules, tells their parents most things that happen in their day, knows what to do in an emergency, doesn't get overwhelmed in a crisis situation, and is good at problem solving. So to me this should be the same guidelines a parent follows on when they let their child go to the park on their own. If they cannot leave their child in a home alone, then they should not let their child go out on their own.
Going back to what my Pediatrician said -- If you let your kids go out on their own, you cannot expect them to always make the right decision; and how he put it, you cannot always expect them to make it back home safely. The best solution - be there, but give them freedom.