Our parakeet would bathe by himself, and he loved it.
When I would clean out his cage, I would pour water (gently, slowly) from my
Brita pitcher into his birdbath, through the bars of the cage from up above.
He would perch on the side of the cage between the stream of water and the
birdbath, and position himself under the stream of water, lifting his wings
and flapping them as the water poured over him. He would stand under the
stream for about 20 seconds, fly to another part of the cage for a moment or
two, then fly back to position himself under the stream of water again. He
loved this and would do it every time I'd clean his cage. When his shower
was finished (I would pour water until he stopped perching on the side of
the cage--it would get the counter quite wet

), I would find him later in
the afternoon holding onto his hanging toy with one foot, and dipping his
chest and wings into the water in his "tub", then flapping his wings to get
the water off.
He loved his showers/bath, and I'm surprised when I read of other parakeets
that don't seem to like water or bathing.