I cant answer your question about staying organic or what people do when an issue like this arises. I'm a firm believer in keeping a healthy flock because mine free range most of the time...I use chemicals when needed to keep them healthy and productive. I dont want sickly chickens. I think as a backyard chicken owner, culling (depends what I'm dealing with) is a waste of time and money already spent on a chicken that might have something as simple as lice and can be treated in less than 10 minutes by dusting, problem solved. Perhaps some would cull, I dont know. Chickens would have to be caged to be parasite free, much like a commercial operation and housed as such to avoid contact with wild birds for transmission of lice/mites and other potential problems and no contact with the ground to avoid internal parasites.
Organic as a general term could also mean the type of feed that they are fed, not necessarily the use of chemicals to treat parasites etc... I also believe that the environment where you live plays a huge role in determining if organic is the way to go or not. I've even heard the term "free range" used as "organic" nowadays. I guess it has many definitions.
Maybe some of the organic BYC'ers will chime in as well.