Parrots love our company, despite how it may seem otherwise. Their biting is quite often just a show of attention, an interaction that's normal for them and would be well received by another bird of their species. When the biting happens, stop everything. Remove the bird from your presence, be very quiet about it and give them 1 to 2 minutes of silence, no eye contact, no interaction, etc.. No longer than 5 minutes - otherwise they forget what they did wrong.
Then pick the bird back up, back to you and have some pleasant time.
You have to be consistent with this method. You can tell them "no biting" or "don't bite" in a firm voice immediately after it happens and as you're about to "shun" them for the 5-10 minutes (no longer than this or it may be ineffective).
When the bird bites, it's getting feedback that it likes and that's why they continues. It might be out of habit, giving them a sense of security in that this action results in that action (or sound) from you. You just have to change the result in order to change the action.
Give your bird time to adjust to a new surrounding and the routine of your home. Always position the cage against at least one wall (in the corner of a room with a view out of a window is ideal).
Always make sure the bird has a shady/cool spot to retreat to in their cage. Since many people work during the day, they aren't aware of how the sun comes in through the window hour to hour, season to season. For this reason, draping an opaque (light blocking) fabric over what you think is already the ‘shady' part of the cage, provides extra assurance. The fresh water supply should also be kept here so it doesn't heat up or become a haven for bacteria.
Once your bird is calm and the home is relatively quiet, approach their cage and just start talking in pleasant tones, or even whisper. It doesn't matter what you say, just talk about your day or what kind of relationship you hope to have with the bird. A particularly ‘problem' macaw came to me so scared of people that she made me scared of her! Every evening, just as she settled in for sleep (but not waking her up; it's not a good idea to disturb a bird's night's sleep) I'd go up close to her cage and whisper how ok she was and would always be. I'd put my head down against the bars and my hands up against them, but lower than she was (height is an important power position for birds). When she'd stretch a wing, I'd stretch an arm (held crooked at the elbow, like a ‘wing').
In my case, I'd leave my head up against the cage even as she struck. There's a couple reasons: 1. I didn't care if she pulled some hair out and 2. Strikes to the skull aren't really painful. They seem to pull back rather than want to go through hair.
If the bird gets overly upset, withdraw and leave it alone. There's another day and many years to come. Time is something you have.
It only took a few days before this macaw made a half-hearted lunge at my hand against her cage and I left it there. She grasped the bars, not my fingers and I was able to stroke the top of her beak, keeping my fingers up over the beak, just under the nares (nostrils) and away from the ‘business part' of the mouth. This was a big step!
When it comes time to take them out of the cage, if you're really concerned about bites, we've used layers of old socks on our arms, hidden by our shirt sleeves so the bird doesn't get used to seeing the socks. To effect the layers, cut the toe end out of old/clean socks and pull them up over your arm, then pulling your regular sleeve over these layers.
Tuck your closed hand down and offer just your arm to the bird with the command (in a normal, but firm voice) to ‘step up'. Put the arm up to the bird's feet (just above them) at the breast and it should naturally step up onto your arm.
If you get bitten (the socks should prevent it from really hurting and reacting), bring the bird to just below your eye level and looking right at them, firmly tell them ‘no biting', ‘don't bite' or ‘be gentle' (whatever phrase you use should be the one you maintain). Transfer the bird to a day perch, somewhere near you as you watch t.v., work on the computer, read a book, whatever it is you normally do. Make sure there are some chewy toys for this perch and if possible, a food and water bowl. If the bird won't be out more than a couple hours and you're sure it ate and drank prior to coming out, it's probably ok if the food/water bowls aren't right at this perch (however, a chewy toy is always necessary to keep beak conditioned and relieve stress). Offer a some grapes, berries, pieces of melon or other treats like this that will also provide hydration.
Talk to the bird, look at the bird, interact with it and get close. Offer your arm now and then just as a ‘step up' exercise, but don't stress it. When it's obvious that the bird doesn't want to do this anymore, don't push.
Eventually you'll find there's no biting happening when you reach in to take them out of the cage. It won't be long before you're doing this without the ‘socks'. You may still get nipped now and then, but the ‘no biting' instruction should curb it.
One thing I keep reminding myself is that as much as it may hurt (this bird I'm interacting with is one of the larger macaws, known for the most powerful beaks in the bird world!), I'll heal. The bird, however, may not. They are forever in captivity by us. We can leave the house, find amusements and distractions, hobbies and interests. These birds are completely at our mercy. Their day, indeed, their lives, are only as full and rich as we provide.
If that doesn't urge our patience and compassion, nothing will.