The context of the piece is what determines how one should interpret modes, so, the question is a little like an M.C. Esher drawing: circular, like a snake eating its own tail.
I'd like to clarify a couple things from your question:
b) major is Ionian, minor is Eolian: C-lydian's relative minor is also
Aminor and parallel minor is Cminor
Natural minor is Aeolian. C-Lydian contains an F#, which is not found in A natural minor. In fact, it is only found in the ascending melodic minor scale. (For the scope of this question, we'll only talk about the three most common minors.) The parallel to A minor is A major, not C minor. Parallel major / minor refers to the same tonal center but altering the scale. For example:
Parallel minor of C major is C minor
Parallel major of F minor is F major
c) a parallel/relative of a mode can be ANY major/minor MODE, not only
This statement is incorrect. Sure, related modes can be respelled and re-contextualized, but you can't go around wantonly naming things; it just doesn't work that way.
Start macroscopically (big picture) and then work your way inwards to the details. Right now it would appear as if you're attempting the reverse: identifying details and then trying to justify a derived context.
Start with the key / pitches used, look at the harmonic pacing and progression, and then derive what modes seem most appropriate given the environment in which they operate.
So, to answer your question about how to interpret a mode in the context of a given piece, my advice would be to interpret it in a way that is logical, that presents a harmonic progression (if the given piece contains harmonic progression) and that most clearly articulates / describes what is going on in the music theoretically. Just because different scales contain the same notes (like D dorian and B locrian) doesn't mean that each scale is completely interchangeable.
Modes are treated differently and are used for different reasons - especially depending on the time period of the piece you're studying.
Hope that helps.