A penny newer than 1982 has a very thin copper plating over nearly-pure zinc.
I don't know whether one "new" penny would ever oxidize fast enough to release enough zinc (or copper) to affect birds drinking from a bird bath, but that article says that zinc is more toxic to parrots than copper is.
If the water is pretty acid, metals could oxidize even though relatively little oxygen dissolves in water. The acidity (H+) can react with metals to release hydrogen (H2) and some metal ions. That's another reason to dislike acid rain!
(I think the article was addressing the risk that a parrot could nibble or scrape actual bits of metal into its gizzard where very acid conditions would rapidly dissolve the bits and release a significant amount of metal ions.)
A "new" penny with scratches through the copper would expose some zinc to the water. If the water was a little acid, a little zinc could be oxidized ... but it seems unlikely to be enough to bother a bird.
BTW: if there is a big enough scratch to expose a good amount of zinc, the zinc would tend to dissolve before the copper, because zinc is more electronegative. Like inside-out galvanizing.
