Well, the answer isn’t any different to when you asked it 14 hours ago 
Why would you expect E maj or A maj to appear in the parallel modes of C ionian? They don’t appear because they don’t exist in any of those modes. With respect to a C maj scale, they are secondary chords, so as a simple answer to your question as to whether or not this display with its current intent will show E maj or A maj, the answer is, as it was before, No.
You could request that here as a new function, and see if you get a response from the team, but you asked this in Jan 22, May 22 (twice), and Sep 22 and got no response, so a priori there doesn’t seem to be much consensus for it.
In each of those cases you included a clip of the Band in a Box Chord Builder, on the basis that it was a much more efficient way to build a progression (not the same thing at all as the modal display). I assume that these previous references to ‘common chords’ is what you are looking for. That’s fine, but it’s not the same as the modal display, so IMHO, that is not a starting point for a BIAB type function.
Personally, I don’t mind if they add the BIAB function or not, but as an ex-user of Band in a Box (which many people here won’t be), I have to admit that I didn’t find it any more efficient than Scaler , but that will depend on your own personal workflow.
BTW, I did point out that you can get the information you seek from the ‘Secondary’ function on the Mod page by looking at the modulation pathways.