The primary problem with doing this on parrots is because they will more than likely ingest at least a taste of it, if not the whole thing! Obviously your snake will not take a bite out of a branch. Of course you want to stay away from the carniferous woods such as cedar and pine because the oils from these wood are toxic to our little friends! Sometimes certain fruit woods are toxic as well, so I would stay away from the plum. Stick with your hardwoods such as maple, poplar (aspen), oak, etc.
The problem with doing this though is not fear of ingestion but parasites when talking about reptiles. If you do this, be very sure that you treat the branch accordingly so that all foreign parasites are killed off!
Different people have different theories on how to rid of them. Some will bake the object at high temps, but not extreme enough to burn! Other use a bleach solution and make sure they are very careful to get in all the nooks and crannies! When you use the bleach solution also make sure you let it dry several days to remove any residual bleach. Chlorine bleach evaporates with 24 - 48 hours.
Do a search here on the forums and on the web for suggestions and usages for amounts and temps. I personally don't use branches or anything like that so I don't know the exacts, unfortunitly. This has been discussed here several times though...use that SEARCH feature, it's wonderful!
Others will hopefully pipe in here and let you know some more opinions.
Quigs