Wow, just had a quick look, is this meant for parrots or all birds in general? Bracken fern, use it for a nesting bunch for my painteds, parsely is listed as toxic in "plants" then safe in "vegetables". Cant follow it, althought the obvious do stand out, lantana, oleander etc. In hindsight it is probably meant more for the "chewers", aka hookbills but still makes interesting reading as I'm sure there is a definite overlap.
Bracken Fern is famous for containing a whole cocktail of poisons including, from memory, cyanaide and various interesting alkaloids. I assume the birds would need to ingest it when fresh and green in order to get the full effect, and that it aught to be relatively inert once it's dried out. So finches are unlikely to be troubled by it. Even if it's growing in the aviary, it would seem to be too tough and unpalatable for them to bother with. Parrots, on the other hand, with their obsession with chewing stuff, would be more of a worry.