I think that's what I said but probably wrongly worded. C natural minor (C aeolian) is the 6th degree of Eb major. Both tonalities are siblings, Eb is the ascending parallel of C, so C is the descending parallel of Eb. When playing C ionian and shifting to C aeolian (what most would say going from C major to C minor) you're "false modulating" from C to Eb. You can improvise on both chord changes (any chord of any of both "tonalities") just going from C major to C minor pentatonic, or even just playing C major pentatonic. Boring but effective. Once you learn to look at single note changes from ona tonality and its modes to another, improvising and harmonizing gets a lot easier and quicker if you pretend to improvise on the fly.
A minor (A aeolian) is just the 6th degree of C major (ionian) so both are into the very same scale and tonality without any note change. You're right, A is the relative minor to C and C is the relative minor to Eb, which is why they're intertwined tonalities. If you assume that A minor (aeolian) is just a mode of C major, both are the very same tonality which is C. Now think of A major (ionian), this tonality despite having very different notes to C major, is its descending parallel. This leads us to another interesting finding, A and Eb are also connected and while not politonal tonalities (sharing the very same pentatonic scale) they share chords and dominants, which is useful to suggest without playing.