Im only guessing here, but Blackbirds swallow berries whole. Parrots bite off bits before swallowing. Its normally the seeds inside the berry that are poisonous, if a parrots' sharp beak breaks the seed coat then there may be a problem of poisoning.
Like I said though Im only presuming to know the answer, I personally wouldn't feed them to my parrot though (That I haven't got :-\)