Iāve gotten into oils recently. Hereās the tea:
No, no oils are safe. But there are safe ways to use them. No, you cannot use them to fragrance a room. Thatās a no go.  Anything atomizing or highly concentrated is a big no-no.  
However, there are ways around it. linen sprays are a good example. You can make blends intended for use on your bed pillow. I created a fantastic one that I spray on my pillow every night before bed   The mix dilution is significantly weaker than most any essential oil blends of any kind will make - 2% compared to burning the 100% pure oil  or an atomizing room diffuser, or even read diffusers at 30 to 40% concentration.. If your birds rarely go in your room, you are perfectly safe with this here.  But again, donāt use it to fragrance the entire room because smelling it alone is not the criteria for safety with our birds. Keep it to the pillow.  
Also, on reed diffusers, you can have these,. But purpose is important: to keep it contained.  My use case: a diffuser at my home desk (I work from home) I can smell, but canāt be smelled more than a couple feet away. The key is keeping it sufficiently weak concentration and using 2 reeds instead of 5+. This keeps my bird, in a cage 10 feet away from my desk, safe. The room is also very open, and I have to run a fan just to keep myself cool so thereās a lot of ventilation going on.
You cannot use oil burners, or atomizer.s. But if youāre making your own blends in weak enough concentrations with a sufficient space and ventilation buffer between you and a bird, that should be sufficient risk mitigation.
Thatās the important thing here. Risk mitigation. You can go the route of absolute  abstention, something youāll often get on forums like this.  but often if you have the capability, risk management gets you pretty darn close to what abstention gets you. 
Keep it contained to a small area.  Well, ventilated and only fragrance within a couple feet of the fragrance source.