I agree, it could be several different things. I suggest you call your county extension agent, in the phone book under county government, and see what it would take to get a necropsy or a fecal sample tested. In some states this is prety easy and inexpensive, but in some it is neither easy nor inexpensive. But at least you would know what you are dealing with.
When I first got chickens in this location, I called my county extension agent, who connected me with a professor at the University of Arkansas that teaches chicken diseases among other things. He also is a chicken breeder and has won some nice prizes from showing chickens. He is also on the regional task force that investigates chicken disease outbreaks. Some people on this forum know who I am talking about. I consider him as qualified as anybody to give advice on chicken diseases, whether a backyard flock or a commercial operation.
In our discussions about chicken diseases in this area and how I manage my chickens, I mentioned that I planned to eat all my chickens before they reached 3 years of age. His comment was something like, then there are some diseases we won't need to discuss. I was not intelligent enough or quick enough to ask why and I don't know which diseases he was thinking about. I could make guesses but I won't. I just don't know why. But it does sound like older chickens might be more susceptible to some diseases than younger chickens.
But 3 years old is not that old for a chicken. That should not be happening happening to yours. To me, it sounds more like a disease than worms or lice, but I try to never totally rule anything out. Check them for mites after dark in case it is roost mites. It is possible that mites, lice, or worms are weakening them enough for something else that would normally not bother them to now kill them. And I would not rule out Coccidiosis. Older chickens often get hit pretty hard by a new strain of of the protozoa that causes Cocci, and all it takes is for a wild bird that is infected to void itself in their area for them to get it.