I've heard of birds getting impacted or occluded by threads of materials, bedding and even grit but I've never heard of parrots dying because of wood splinters although I would assume it could happen. All my larger species chew wood. It's what they like to do best. But I don't think any of them swallows it - at least, I've never seen one doing it or found anything in their poop.
It is important, though, to make sure the wood is completely untreated and non-toxic and, if colored, that safe dyes were used in the process.





