The parallax knob does in fact put the sight picture in the same plane as the reticle. I guess I was wrong, by listing all the assumptions you made, because as you explained, it all goes back to one assumption. that your eyepiece is adjusted correctly.
I know I adjust all of my scopes to me, however if I'm using a friends rifle, I don't mess with his settings, and vice versa. So that is why I stand by my initial statement that fully focused and parallax free isn't the same thing and you can have one without the other. That is why if its an important shot, I always check the parallax (moving head side to side) as opposed to relying on the focus.
To the others, worrying about their scopes not having parallax. Depending on the application and magnification , its not a big deal. Parallax control isn't as crucial for low magnification optics. If I remember correctly, the effect is multiplied by the amount of magnification. So you will have 8 times the parallax error at 24x than at 3x.
Lastly, all the shortcomings of a scope without these features can be overcome by excellent marksmen fundamentals. Be 100% consistent with your cheek weld and body position and parallax is a non issue. The correction is a crutch for when your not 100% consistent. There are iron sights shooters that have no such thing as parallax correction and can out shoot most of us with our fancy scopes....