This is not well researched as far as I can tell. I've heard natural essential oils implicated as well. Anything with a strong odor is potentially going to irritate their respiratory systems. On the other hand, it does not seem to be like PTFE fumes where the entire house can be affected without you noticing anything with immediately fatal results. Ventilation and proximity seem to be key with scented candles and air fresheners. And I think you would probably notice some discomfort, if you were observant, long before there was any real damage done. But I am not a vet, so I am just saying how I view it, not how anyone else should view it. I have no verified facts and I hate that!
My personal approach is that I don't use anything with a strong scent in the room where the birds live. No candles, no air fresheners, no cleaning products beyond poop-off and mild soap and water, and if there is any kind of work being done on the house I am very careful, close off their room, and open the window wide for ventilation.
On the other hand, our house is fairly large and I do use scented candles and even plug in air fresheners in other parts of the house. I try to keep any strong scents away from the birds and vice versa, but I was using Yankee Candle Company products before I had any idea they might be harmful and as there were no adverse effects up to that point, I further restricted my use, but did not completely stop. I stick to fairly mild scents like vanilla and lavender in the main part of the house, and mostly burn scented candles in either the bedroom or my studio, which has potentially worse things in the air than candle scent, so it is located relatively far from the bird's room and I open windows and close doors appropriately depending on what I'm doing. The birds are not allowed in the studio area if I've been doing any work in there.
We also have a good vent fan in the kitchen and we use it religiously. Strong odors or smoke from cooking food are also potentially problematic! Zataran's boil makes my husband sneeze and his eyes water if we don't use the fan, and that strikes me as more likely to bother the birds than a vanilla scent which has no such irritant effect. You pays your money and you takes your chances IMO.
YMMV and I hate to operate in a veil of ignorance, but we just don't know what components are potentially harmful. I avoid things like Febreeze that suggest they permeate the air and "clean" it, but I think this is one of those areas where you either take a very extreme standpoint or else you take a small amount of risk and make your own choice as to what you are comfortable with. No one really knows for sure what is OK and what isn't, only what hasn't caused them, personally, problems yet or what has! And as has been said famously "The plural of anecdote is not proof". There is no proof here. Frustrating.