Why is the paralaxing shaky? (video examples)

Your platforms are moving using velocity. That means their position is being updated on FixedUpdate(), which is called ~50 times per second by default.

Your background is being repositioned on Update(), which is called once per rendered frame, which could be quite a lot more often if your framerate is high.

The difference between these two update rates can create judder like you're seeing.

For example, if you render at 30 fps with a physics step of 50 Hz, there are 5 physics updates for every 3 rendered frames; that means two thirds of your frames have 2 physics steps between them, but the odd frame out gets only one physics step (so the platforms seem to move less in that one frame compared to the others, making their movement look shaky).

One way to deal with it is to enable interpolation on your platform rigidbodies. What this does is offset the rendered transform in-between physics updates to give a smoother appearance. Interpolation: Extrapolate should predict correctly if your platforms are always moving at a constant velocity as shown here.

Switching both background & platforms to update the same way (either both in Update or both in FixedUpdate) will also fix any discrepancy between them, but might not be practical depending on their gameplay needs.

A more reliable fix is to move your camera instead of every co-moving object independently - fewer moving parts, fewer things can go wrong, and there's less data to interpolate. ;)

How is the build performance though? I find everything is more fluid in the builds, from my experience. I thought this was normal because the editor is still running.