With a Kagi search I found the following info. On a side note, at least concerning this article, the VCA seems behind the times and recommends discouraging masterbation in birds, they also do not mention female bird masturbation. I assume that female birds masturbate in the same way because all birds have the same external genitalia.

From the VCA animal hospital

Male birds may be seen rubbing their cloaca or vent (the underside of the tail) on a favorite toy, perch, or mirror, but can also perform this behavior on the hands, arms, or shoulders of a person.