Green’s Dictionary of Slang

cockatoo n.¹

the penis.

1653

Urquhart (trans.) Gargantua and Pantagruel (1927) I Bk I 44: And some of the other women would give these names, my Roger, my cockatoo, my nimble-wimble, bush-beater, claw-buttock, evesdropper, pick-lock, pioneer, bully-ruffin, smell-smock, trouble-gusset, my lusty live sausage.