I seem to recall a discussion of quite a lot of "polka music" being often in pretty strict thirds. It's apparently pretty common in some such music for the "first harmony" part to move closely parallel and a third below the "melody" part. In some polkas, it's the two harmony voices that move in strict thirds with each other, usually below the melody.

The "bi-modal" terminology would suggest that both voices are in the same key, but starting on different notes and moving in parallel they would be in two different "modes" within the key.

The objection to the sound shouldn't be that it's strange, but more likely it's pretty boring when one is used to modern harmony in which chords change in regular fashion through a sequence of "strong" chords consonant with the melody(?).

John