Well, my birds are not dead yet. I was forewarned about candles-but I love candles. So, I burn them in the next room away from my birds. As long as your bird isn't shut up in a room with no ventilation with a burning fragrant candle, I would say ok. Also, beware of any accidents that could result from a burning candle and a flighted bird. Or, even if you keep their wings clipped they are grown in and before you know flitting from here to there, so careful of the flame.
There is a bird safe candle on the market, Mia Bella is the brand name. You can find them on line. They are a bit expensive but smell just like what it says it is. They seem to last a while also. I have burnt them in the same room as my birds. The place that recommended them raises birds and has a bird store. They have burnt them around the birds, probably 100 or so. Everyone is fine.