Of course I'm not talking about an equal mixture of water and Dettol, just a cap full into the bucket. That's how I do it with my dogs and they've never had a problem. Speaking of dogs, I don't have bird shampoo. Can I use my dog shampoo on my cockatoo? The reason why I want to use Dettol is because I suspect my cockatoo has mites.
What should I use?
Never use anything but plain water when bathing your bird, even shampoo's marketed for birds are not good.
Birds have a preen gland called the uropygial gland which secretes oils to keep feathers and skin healthy. Shampoos can remove these oils, leave soapy residues and dry their skin (which can ultimately lead to feather destructive behaviors).
Parrots should have access to a bath at minimum 3 times a week.
What do I do if my parrot has mites?
Your first step is to go to your avian veterinarian to rule out mites, once a diagnosis is completed then the appropriate medication will be given. Typically for mites they are given an oral anti-parasitic.
If mites are the cause you will need to heavily disinfect their cages, I would throw out anything that is difficult to clean (i.e cotton ropes) to prevent reinfection.