There are different ways to thing of this. Minor to parallel major is a bit of a weird move sometimes. Often people think or use the opposite, ie, Major to parallel Minor.
That said, it depends on what style, what sort of chord inversions and/or voice leading you are comfortable with--> or need to get into the tool kit!
A couple of ideas. First, Dminor is the relative minor of Fmajor, so you can change perspective, so to speak, by thinking of going from Fmajor to Dmajor. How about going Fmajor to Bminor(relative minor of Dmajor)? That's weird, and gives you a sense of the whole thing being a bit trickier than it appears.
Voice leading is a good way to accomplish the original question. (You did it transitioning back.) The good, old standard a la Freebird: b3 b7 (new chord in your case) can work well.
ie-> Dmin / / /|F / C / |to-->>| Dmaj / / / ||-->>to whatever you come up with for Dmajor
You don't have to (and maybe shouldn't, depending on what you're aiming for) just use a big, clumsy bar chord movement. Voice lead something like the following:
F = Fmaj/A or spelled [A F C]
C = Cmaj/G or spelled [G E C]
and then to Dmajor: Dmaj/F# -> spelled [F# D A D]
Something w/ a bit more movement and flash:
Dmin Dmin/C GmajB | Amin Cmaj/G -> Dmaj/F#
the above would be spelled (bottom to top)
- Dmin = D A F
- Dmin/C = C D A F (you could also drop the D and think of this chord as either Dmin/C or as F spelled -> C A F
- G/B = B G D G
then move to next part:
- Amin = A E C
- Cmaj/G = G E C
- Dmaj/F# = F# D A D
That moving from Minor to Major can have a sort of "majestic" sound.
Wow. I've been playing w/ these key/chord changes all day. There are TONS of fun ways to go about this!
- Quick hint for now: I mentioned thinking of Fmajor in place of Dmin (relative minor).
Now, go a step further--> Use the parallel minor of Fmajor --> Fminor, or Abmajor.
What I'm coming up with for myself is weaving together inversions of Abmajor and moving to Dmajor.
This is what I used for Dminor. Pretty basic, just wanted to get the sound
|| Dmin / C / | Bb / F / ||
Then moving into Fminor I used these chords:
| Fmin/Ab / E/G# / | Db/Ab //// | F#maj / F+#9 | Emin7b5 / Eb/G / ||
What I messed w/ from there went (in Dmajor):
|D / / / |E/G# /// |G /// |F#min ///|
|F#min///|E / / / |G /// |D / / / |
|D / Amin Fmaj7 | C / Bb / ||
From the Bb I went back to Dminor:
-->|| Dmin ||
I hope that makes sense. It's just a sketch, so the exact number of measures, etc isn't proper sounding yet. Not sure how you would make this Dixie style, but very fun.