I think some of these lists are generic; that the researcher did not study what species is made sick by what plant. For example, we all know that poison ivy leaves a rash if humans touch it (or, for some, just get close to it), and would certainly cause a bad reaction if they ate it. Yet poison ivy berries are one of the most valuable winter foods for the wild birds here in Ohio. Polkberries are also eaten by wild birds here, but are very poisonous to people.
As for parsley (and some other valuable nutritional herbs on the list), I have read many articles by respected parrot experts who use and recommend it and some other herbs on the list. Parsley does have some medicinal value for people, but it is the quantity and frequency that makes it a medicine. Allowing a bird to munch on parsley isn't going to poison him.
I prefer to work with lists that have the Latin names with the plants because of the confusion possible from going by the common names. Pigweed, for example, is a name for different plants in different parts of the country. I can think of three different plants that go by the name "myrtle."
That said, for goodness sake, please don't think the list is useless! Use is as a valuable guide, and triple check any food or chew item you want to give your bird!