In the Parsons Code section, if I'm reading correctly, the chords are conceived along an up-down axis. However chords do not really have that relation. Going from 1maj to 5maj can be up a fifth or down a fourth.
To illustrate this, first listen to the default 1645 progression on rhythatom. Then change the octave of row 2 (4maj chord) from +0 to +1. Can you hear its similar role in the progression, even though it's now stressed by being up an octave?
I talked about this with my friend Lenard, because I was confused why the Parsons code would be different for the same chord progression.
He then explained that I should move the key, and when I transposed the MIDI table, the Parsons codes all became the same.
I then asked him, is it possible to move up an octave for just one part of a chord progression? He said yes, it's possible, but people don't do that often. Normally people wait until the end of a chord progression to change octave.
Talking about going up or down an octave here is just nomenclature. Musically speaking, changes in a chord progression don't go up or down - this is because roman numerals refer to pitch classes (like C, E, G), not to specific pitches (like C4, E5, G3).
The matter of which notes to play for a given chord is called "voicing", and it's orthogonal to anything mentioned in your article.
To illustrate this, first listen to the default 1645 progression on rhythatom. Then change the octave of row 2 (4maj chord) from +0 to +1. Can you hear its similar role in the progression, even though it's now stressed by being up an octave?