How do you describe the Everly Brothers' harmony in musical terms?

[BernDog]

Moving in parallel thirds could be their defining sound (never analyzed it myself, so, i'm just going with that for the sake of argument). That would mean that it would be a common device in their harmonies. As you found out, though, it becomes ridiculous when used exclusively, and probably wouldn't consistently even fit the chord structure. Something similar could be said about the horn sound used by Chicago being defined as trumpet and trombone moving parallelly in octaves. They don't always, though.

Edit: regarding the thirds, you also need to make sure the thirds are thirds within the key, not just the interval of a major third. Sometimes it would be major thirds, sometimes minor.