Biting is known as a self-reinforcing behavior. What this means is that when a parrot bites, even if you don't yell out or grimace or show any response whatsoever, the parrot will enjoy that bite and therefore be more likely to bite again.
Of course, if you DO make a fuss, this is extra entertainment for your parrot, so you are only providing bonus reinforcements for the biting behavior.
Parrots bite when they are either over-excited or when they are fearful. So, what you need to do is to learn how to read your bird's body language so that you can recognize when your parrot is in either of these states. Whenever your bird's body language is telling you that he/she is over-excited or fearful, you need to make sure that you are not within biting distance of your parrot, so that you do not get bitten.
Things to watch out for include body movements (whether the parrot is advancing or retreating), body posture and body stance, any neck craning, beak opening, eyes pinning, how the wings are held in relation to the body, how the feathers are held (especially the feathers on the nape), whether a foot is raised, whether the toes are clawing, etc.
Once you know how to only be within biting distance of your parrot when he/she is in a calm and relaxed state, then you can start using positive reinforcement to train your parrot to target. When your parrot is well-trained in targetting, you can target him/her whenever he/she starts to climb up your arm, and re-target him/her back onto your hand/wrist. If at any stage, your parrot's body language shows you that he/she is starting to get over-excited and/or fearful, then it is time for time-out so that your parrot can relax and calm down.
You can also use positive reinforcement whenever your parrot is well-behaved on your hand/wrist, and reward him/her with his/her favorite treats. This will help your parrot think of your hand/wrist as the favorite part of your body to perch on.
All this is easier said than done. I still need to work on preventing Chilli from climbing onto my shoulders. My dear old dad, unbeknownst to me, let Chilli perch on his shoulders all the time AND fed her biscuits while she was there. Is it any wonder that Chilli always wants to go straight up to the shoulder?!? It's going to take a L-O-N-G time (if ever) before this behavior is extinguished!
Hope this helps,
Enjru