Right, like so many other things in music theory, the name can be misleading.
Ok, now you hit the point. In my view, keeping the same keynote pressuposes that when you replace one chord by another from the parallel scale (in the same degree) the chord function in the progression remains unchanged. I'm not sure if this is correct.
I've been searching (and found) lots of different sites and books and any of them could answer this question. They all just mention the substitution with a few examples and nothing more.
I tried to understand this by comparing both, the CMaj and C Natural minor scales:
Modal Interchange.JPG
During my search, I concluded from several examples that only the only chords that could be replaced by chords from the parallel scale were the I, IIm, IV and V (because all the examples were just dealing with these 4 chords).
So, by looking at the table above, I concluded that this was because they were the only chords that have the same root. All the others, in the minor scale the root is flattened by a half-step.
By other hand, and considering the diatonic chord substitution, if we look to the fundamental chord structure (triads), the I, IIm, IV and V all have two tones in common (which doesn't happen with the other chords).
So, I found these two as possible reasons to explain why these chords are interchangeable.
But after that I've found other examples with chords other than those.
So, this "theory" of mine, although plausible, doesn't explain everything unless we replace the IIIm, for example, by a first inversion of the bIII. But would be replacing a Em7 chord by a G based chord (in this case something like Gmb6). As I said before, I don't know if changing from an E based into a G based chord will change the function in the progression.
These are my doubts for now.