The movement of the sun would produce the same stellar parallax.
No it wouldn't. No matter how the sun moved it would not cause near-by stars to move in relation to more distant stars.
You can see how it would work in the image. If you want, I'll try to create a geometrically sound geocentric framework for the parallax of the stars. It might take me some time though.
All you have to do is switch the Earth and Sun in your diagram. Notice that the star would remain stationary above the middle object as the outer object orbits around it.
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