It is not neatly broken down into new world parrots and old world parrots, because not all old world parrots are closely related and not all produce powder down. Vasas lack powder down and should be managed like New World parrots. Amazons are South American but they produce their own sort of powder dander which may be harmful just the same in close confines without good ventilation.
With sufficient ventilation, you are unlikely to have issues, especially with smaller species like cockatiels, kept in the same space as non-dusty parrot species.