I just wanted to bring up what I consider to the difference between taming and training. I'm not really interested in the dictionary definition but more about what techniques apply to each.
I consider training to be the process of teaching a cued behavior that requires the parrot to expend some kind of effort. Whether it's picking its foot up, saying a word, etc, these are all things where the parrot has to expend energy to make it happen. I believe this kinds of behaviors can solely be taught using positive reinforcement. There has to be a motive. It does not absolutely have to be treats but food works best. However, whether its attention, a toy, or food, the parrot has to have something to gain in return for a trained behavior.
Now taming on the other hand I would consider to be a reduction of fear, aggression, or other undesired behavior that isn't necessarily reinforcing to the parrot. If the parrot is fearful of being touched, it can be tamed to accept touch. The parrot doesn't have to do anything or expend any energy, it merely has to resist it happening. So for taming, I believe that a combination of positive and negative reinforcement are effective. The negative reinforcement is that you only do the undesired action to a certain extent (while slightly pushing your parrot's threshold further and further with successive attempts), stop, and then eventually retreat. The positive reinforcement aspect is similar to training. By using multiple methods simultaneously this way, you can greatly expedite the taming processes.
The difference long term difference between training and taming is that training requires both practice and a continuation of reinforcement. Otherwise extinction will occur and the behavior will be forgotten. Taming on the other hand no longer requires reinforcement once it has been taught. The parrot does not have to make any effort to be grabbed, handled, wings opened, etc. It merely has to not resist. Once these actions have successfully been tamed, just some occasional maintenance is required to preserve the tameness but positive reinforcement is no longer necessary. So that's just my take on training vs. taming.