Maybe I'm missing something.
Which is just another way of saying that every major chord has a relative minor
You are most definitely correct that each Major house is also it's
relative minor house, but..........
We are discussing
Parallel Major and Minor, in this case switching between the
A Major and
A minor sounds.
In your example it would be like starting in
Em and going to
E Major, rather than staying in the same house and changing the root to
G (Major).
(Summary of the discussion so far)
1) Move the same Box up or down
3 frets. (Ex: Am Box 1 down 3 frets = A Major).
2) Or staying in position and combining two consecutive boxes.
(Ex: Am Box1 and A Major Box 2 = Composite, as in the "Stitch" video which combines minor Box 3 and Major Box 4, which he calls the BB Box. This is equivalent to taking the Major house, bending the 2 up to b3 and 3 and adding the b5 = 1 2 b3 3 4 b5 5 6. He doesn't mention the b7, but you could add that in as well. You then have all notes except b2,b6,7.)
3) But we also have the unique case where minor box 2 has the same shape (house) as Major Box 4 (on the 3 top strings), with one exception, the Major 3rd which is 1 fret lower on the 1st string. However, we can get this note by bending the 2. So in this case we can move the minor house up
2 frets to Major and have the exact same note as the root in each. Played with 2nd finger in the minor and the 1st finger in the Major. These 2 "houses" actually
overlap.
This is also addressed in "Soloing Without Scales" by moving the "4 Note Solo Pattern"
2 frets, retaining the same root. It's just the house without the "roof".
It's worth it to compare the "Killer Blues" Solo, the BB bonus in BGU and the BB solo from "Solos From The Masters." Griff uses a different one of the 3 approaches in each of these 3 solos.
If for some reason you wanted to go from A Major to it's
relative minor, F#m or from A minor to its'
relative Major, C your approach would be correct; play the same box from different roots.
As opposed to playing the same root in different boxes =
Parallel.