I don't know about giving it to budgies, but I know that the orange side of the peel has some very strong oil in it, that is used only in tiny amounts as a flavoring for humans, whereas the thick, white side of orange peel is actually 'Richly endowed' with bioflavonoids. I wouldn't think that the white side would harm a budgie, & may even be very good for it. However, I'd with-hold the orange side. I used to have a budgie, and I regularly gave her fresh twigs from a eucalytus tree. It seemed to enjoy debarking the twigs! There are strong oils in the leaves, though, that might not be so good for a budgie ...
Twigs of fruit & nut trees may be safest, but others may be good, too.
What Birds Can Teach Us ...
http://watchtower.org/library/w/2003/6/15/article_01.htm
Pets--How Do You View Them? :
~ Animals--A Gift From God
~ Pets--Keeping a Balanced View of Them
~ Animals--A Pleasure Forever!
http://watchtower.org/library/g/2004/2/22a/article_01.htm
(These URLs will likely be modified soon, after which the on-site Advanced Search at http://watchtower.org/search/search_e.htm will find their new URLs.)