The attached is my reproduction of the ubiquitous Edmund Fellowes/Stainer & Bell edition, using more readable note values and barlines (if recent experience is anything to go on). The passage you reference does use parallel fourths, but I read them as a sort of modified version of fauxbourdon, the movement of parallel triads in first inversion. (In this case, the bass takes the middle note of the chord.) This is indeed characteristic of English polyphony (if a bit anachronistic by Byrd's time), and, I would argue, unavoidable given the three-voice texture.

The affected measure rendered in strict fauxbourdon would look something like the sample attached.