I think yours is an interesting remark, @Mark_Johnson, which exposes a stance on parallels shared by many authoritative music educators, yet rather different from mine at a fundamental perceptual level.
What you (seem to) feel in a mostly harmonic sense - when chordal compounds shift statically to displaced replicas, through equal movement of every layered voice -, I tend to detect in the interplay of the contrapuntal construct.
In both cases, I think that the somehow “rougher” musical quality of parallelisms reveals itself when the phenomenon repeats, outlining a mirrored pattern between parts that appears to flatten the spaciality of musical movement, in a more rigid plane devoid of intervallic volume, if I can attempt this metaphor.
I never felt that parallel fifths, which I sense as having a certain “electrical” brilliance to them, were musically wrong, neither in aesthetic nor ethical terms, especially when they happen in a complex concertato texture between voices that interact convincingly, with appropriate contrast of movement and rhythm. More than two exposed consecutive fifths can sound elementary instead, like lacking effective voice-leading, if there are unexploited alternatives at hand, unless… the composer is legitimately after their sort of algid and archaic monumentality, for conscious expressive purposes.
The matter of parallel octaves is much more sensitive, of course (I am not talking about doublings, which are fully employable by definition). Octaves can and sometimes have to occur in large musical canvases, if they are necessary to preserve higher formal priorities, like the integrity of a thematic reprise or imitation, a crucial shape in a certain part or some harmonic content of greater importance.
Hidden parallels can be treated, in my opinion, with a much more relaxed attitude in real-world compositions for more than four parts (not tuitional exercises), providing that they not stand out in a wrong expressive way (according to style and language, of course), for example between melody and bass or a couple of voices in strong acoustic relief.
The general criterion stands (avoiding awkward movements with parallel leaps to weak acoustic intervals, in voices that obviously proceed interlocked), but its practical application, once learned its basic rationale, tends unavoidably to become very flexible.
In such a context of implied theoretical and/or stylistic reflection, I believe that a reliable on-board parallels checker would be a useful compass for autonomous orientation, for many Dorico musicians. Its use would not be to instruct the composer as to how handle himself the more or less unavoidable subject of voice-leading, but to help him/her to manage it in a technically and culturally conscious manner.