I'm writing this post because I recently answered a question about how to stop a bird from going to your shoulder on a board that I frequent. Sometimes it's easier to write a post so that I can direct people here in the future, with pictures, instead of retyping everything again!
First, like many parrot-related issues, whether or not to allow a parrot on your shoulder is an individual decision. I've always been a worst-case scenario type of person (a visit from a former smoker whose voice box had been removed -- similar to this person -- in the 3rd grade convinced me never to smoke because I was sure that would happen to me if I did).
When we got Max, we decided that we didn't want her on our shoulders. The main reason was that we didn't want to risk a face bit. Here is a page with pictures of some face bites that have happened. Later, we realized it was a good choice for us for other reasons. It's hard to see body language when the parrot is on your shoulder!
At the parrot rescue where I volunteer, I was in a room with another (now former) volunteer. He had a large macaw, who loved him, on his shoulder. He was walking around and got a little too close to another bird. As a warning, the macaw on his shoulder gave him a nip on his ear that managed to pull a few centimeters of cartilage out of his ear. This confirmed our decision to not have shoulder birds! I've also seen horrible ear lobe destruction by a shouldered senegal (I witnessed the attack) and have seen the results in the form of facial lacerations on at least a dozen people surrendering their birds to the rescue where I volunteer.
However, I realize that thousands of people successfully allow their birds on their shoulders, and I don't believe that one way is necessarily better than the other, assuming that people have thought about the risks they're incurring. Even in our house, Thomas will occasionally place Rocky on his shoulder, and I'll occasionally allow Daphne on mine.
Four of our birds (everyone but Max and Daphne) came to us as shoulder birds. When they got on a hand or arm, they would automatically scamper up to a shoulder. We've managed to break all of them of this habit.
This post is meant to help people who want to keep their bird off of their shoulder.
In my opinion, the two most important components of preventing a bird from going to your shoulder are arm position and persistence. Max was kind enough to stand in as my model for arm position pictures!
Many people hold their parrots in such a way that basically gives the parrot a ramp to their shoulder. Many birds like to be up high and tend to gravitate to the highest position they can. When I hold Max like this, it would be very easy for her to run up to my shoulder.
To prevent a bird from going to your shoulder, it's helpful to arrange the environment to make that as difficult as possible for her. I've shown this position to scores of people with this issue at the rescue, and it can be difficult at first to get used to!
Keep the top part of your arm as close to your body as you can. It's a lot more difficult for a bird to climb up a vertical arm than a ramp! Then, make a "V" at your elbow. The bird is up higher (you can see in this picture that Max is almost as high as she would be if she were on my shoulder). If she starts climbing down my hand (which often won't happen because birds like to be high and that requires climbing down), I have plenty of time to get her to step up on my other hand and replace her where I want her to be.
Another reason some birds like to be on a shoulder is that they feel safer there. This is especially true of parrots with poorly-clipped wings, or anyone with balance issues. For birds like that, I'll keep the top part of my arm as vertical as possible, but I'll bring the bottom part of my arm in. It's almost like I'm cradling the bird with my chest. The bird will often rest their beak against my torso when we move, giving them extra security. Depending on the bird, their condition, and my relationship with him/her, I'll sometimes take my other hand and hold the bird closer to me. I seem to do this especially with large macaws who have foot and/or balance issues.
Persistence is the other key aspect. If you remove a bird 20 times from your shoulder but then get tired of it and allow her to stay there after the 21st attempt, you've just taught her that she'll get her way if she keeps on trying, and it becomes more difficult to extinguish the behavior.