what type of modulation moves to the relative minor of the parallel major? For example....The key of b min moving to the key of g# min. g# minor is the relative minor of B Major which is the parallel major of the song's original key (b min). To add another concept,a D# Major chord is used as the intermediate transition chord between the modulations.
Since you mentioned “Unbreak My Heart” I can say the modulation from Bm to G#m is less significant than the modulation from Bm in the verses to Dm for the choruses, which is the same distance from Bm but in the opposite direction. That is a chromatic mediant modulation as would be Bm to G#m (see my next paragraph). The modulation to Dm is clever. It uses the bVII chord of Bm and turns it into a V in Dm.
The G#m modulation is for an instrumental interlude and actually modulates from the Dm chorus, not from Bm. @Dekkadeci said Bm to G#m is a chromatic mediant modulation. He is right but he didn’t have a musical example and was not aware that the song actually goes from Dm to G#m. Dm to G# is not a chromatic mediant modulation, it’s an extraneous modulation like @Tim mentioned, or a tritone modulation. It gets there with the D# chord you mentioned, which is the V of G#m and also a sub V of Dm. It then returns back to the original key of Bm by using the bVII of G#m as a V to pivot back to Bm, another chromatic mediant.
Here is the basic layout of the entire song:
Intro Bm/Verse Bm/Chorus Dm/Verse Bm/Chorus Dm/instrumental G#m/half verse Bm/Chorus and vamp out Dm
My 'bible' labels it as extraneous modulation - if the key has actually changed. I was considering what John wrote, where one minor chord went straight to another, as described, as in 'Light my Fire'. Which was simply a change from one chord to another, related in a 'parallel' way.
The D♯ is not surprising - being the V of the new key.
You could call it 'moving to the relative minor of the parallel major'.
Or if you must hang a label on it, 'chromatic mediant' (which is generally agreed to cover sub-mediant as well).
The whole 'mediant relationship' idea is a bit artificial anyway. Yes, it's reasonably painless to shift to a triad that has one note in common, and it's reasonably painless to shift to another triad of the same flavour - major or minor. I'd suggest that that's sufficient excuse for 'chromatic mediants' and for a lot of other shifts besides.