I like the idea of the fancy cockatoo being the more important subject of the painting, like "oh hey, who's that Gustave is hanging out with?" "idk, some woman at a piano" "well, lucky her"
The artist worked out of Belgium and France, so this is almost certainly a cockatoo that was either taken from the wild or hatched locally. Most people didn't care much about protecting wildlife back then.
Back then, cool-looking wildlife was often exported worldwide for the amusement of bored rich people. And a large, unusual-looking parrot from the other side of the globe would have made a desirable conversation piece.