The thing that helps explain these harmonic rules best for me is trying to understand at the rootmost level what the goals were of the composers of the era. My impression is that they felt that the major or minor third and their inversions were the "sweetest" interval. Fifths and octaves are used only to enrich the third. All the harmonic rules of the common practice era revolve around treating the third with care. Some rules are to taste, such as doublings, but you'll find thirds doubled less often than fifths, and fifths less often than octaves. This is all because of emphasis of the third.
It made the most sense to me when I bought my clavichord and improvised with and without paying attention to these rules. On such a quiet instrument with a metallic timbre, dissonances are very harsh, and consonances are very thin indeed. However the imperfect consonance of thirds have this remarkably unique character on older keyboard instruments. They cause a greater immediate physicological response in the ear of pleasure. Thus I believe these instruments are the true father of western composition. Fifths in parallel actually sound fantastic as we all know today played by instruments of more complex timbre (piano, organ, electric guitar, etc. etc.). On the old keyboard instruments, they just sounded "flat." Hard to describe in a couple of paragraphs.
Please note I'm an amateur, and these are my own findings and conclusions, not necessarily corroborated by musical academia.
Topic: Parallel fifths explanation.