In fact in any music, parallel anything all the time gets pretty boring, but to imply, as you did in your earlier post that parallel thirds should be avoided is not exactly helpful.
It is not helpful to dismiss classical music theory in the way you do, for all its faults it has stood the test of time and has adapted to changes over time. I found the harmony part of the theory course very helpful, but it should not be followed slavishly. As to the harmonisation of modal tunes, it is the tonic-dominant harmony that does not apply rather than use of parallel anything or contrary motion. In fact if you really want to be "purist" you should not be harmonising modal tunes at all as the modes are about melodic structure, not harmony.
In practice, as you rightly say, it is your ear that has to be the judge. However, an understanding of harmonic principles, whatever source you get it from, does help.
As I play Anglo Concertina (albeit badly - I am only a beginner), then harmonic choices are more limited and when I am trying to work out an accompaniment, I generally ignore most suggested chords as they are not always practical in the particular circumstances.