Here's a puzzle, though: parallel fifths became perhaps the dominant type of chord transition in the 80s and 90s with the rise of hard rock, punk, and grunge, all of which heavily featured the "power chord," i.e. fundamental, fifth, and octave. How can the popularity of the power chord be reconciled with the prohibition against parallel fifths?
The easy answer: parallel fifths have always appeared in low-brow, "popular" music, and they are just a sign of the general incompetence / lack of complexity of such music.
But this answer is not satisfying. If we reject the possibility of widespread poor taste / ignorance, then we should look for a more interesting answer, one which addresses the reasons for the prohibition against parallel fifths, and how or whether power chords violate these reasons.
The exact reason for the prohibition is unclear. Wikipedia gives several:
1. It interferes with the independence of voices in counterpoint.
2. It reduces the number of voices.
3. It throws the key into doubt.
None of these reasons seems especially deep, however: they depend either (1 and 2) on the assumption that one is concerned only with the integrity of one's counterpoint, or (3) ambiguity of key is undesirable. Both doubtful assumptions for harmonic music in general.
Going back to the most influential source, Fux's translator, Alfred Mann, draws our attention to a particularly important passage:
The purpose of harmony is to give pleasure. Pleasure is awakened by variety of sounds. This variety is the result of progression from one interval to another, and progression, finally, is achieved by motion.
"Direct motion" is that here called parallel, and it is in the following discussion of motion which Fux introduces the prohibition against parallel fifths or octaves. These are the perfect consonances. Direct motion between imperfect consonances, the third and sixth, is permitted. The remaining intervals (including the fourth!) are classified as dissonances.
Mann offers an enlightening gloss in a footnote to the quoted passage:
The statements, which introduce the following fundamental rules, may at the same time be considered an explanation of the principles of voice leading which they embody. The "variety of sound" is the basis from which all further rules are derived: first, the prohibition of parallel successions of perfect consonances, as depriving the voices of their independence. . .
Here, we see again the mention of independent voices, but now we are given a reason for the importance of that independence which transcends a mere commitment to x-voice harmony. Rather, we are given an explanation for the function of harmony (to give pleasure) and a theory about how it does this (through variety). Parallel fifths (perfect consonances) do not present enough variety to fully fulfill the function of harmony (while moving imperfect consonances do).
But this explanation sheds light on the popularity of power chords. The parallel power chord arose as a technique on the distorted electric guitar. Distortion adds new (potentially dissonant) overtones to a sound. This increase in overtones makes Fux's imperfect dissonances (e.g. the third, which usually complete a major chord) more dissonant than with a clean sound. This provides a negative explanation for why they are dropped in the power chord.
But Fux via Mann has now given us a positive explanation as well: parallel fifths are permitted as distorted power chords because the distortion adds the necessary degree of imperfection to the otherwise perfect consonance of the fifth / octave, thereby introducing enough variety to allow their successive appearance to give pleasure.
In fact, arguably, this deeper (psychological) analysis of Fux's principles mandates the switch to power chords in the context of distortion. If Fux classifies a clean fourth as a dissonance, he would certainly classify a distorted third as one! If we allow the analysis of intervals into dissonances and consonances to change with the overtones generated by the addition of distortion, then Fux may be interpreted as demanding the abolition of all but the octave and fifth in a distorted context.
[Further motivation for this interpretation can be found in the analysis of musical traditions which use instruments with very different overtone profiles than those encountered in classical Western music, e.g. Javanese Gamelan music.]
Of course, there is an unhappy consequence of this analysis. Music performed on distorted instruments is limited in its harmonic / counterpoint complexity. Of course de facto this is what we observe, but it is striking to discover that it might be a limitation on the cognitive effects of particular timbres, rather than merely a limit on the sophistication of popular musicians. In fact, on this interpretation, it is not a limitation on the side of the musicians (what they can conceive, etc.), but on the side of the listener (what they can hear as pleasurable) which motivates the use of powerchords rather than more complex counterpoint in the various flavors of "hard rock."