All solutions focus on particular pawn position... but there are others too:
All pawns have to have a single possible move. It is obvious that either pawn has to move a tile or two, OR there has to be something blocking it from moving 2 squares - say another pawn. Note that pawn's only move could be also capture (of another pawn most likely)
16 moves are theoretically minimal - but there are many such options. You can see that for example one with white abcd pawns on row 5 and black efgh ones on row 4 also work at same number of moves. Piece blocking pawns is not completely impossible either, though the only solution I found was a king between 2 of his own pawns, with 2 other figures behind those 2 pawns, a space directly behind, then a pawn 2 steps behind. White's pawns remove option of moving forward.
Like this: (capital letters for black, small ones for white):
I haven't analyzed if this is actually possible or breaks at some later point. Let's forget about this one for now because I couldn't see how to use it and rather consider something else:
Let's briefly check how few moves are needed to hammer white's side only: a3 gives rook a single move and takes a square from knight. Great. It would be wonderful if b3 is played to offer bishop a tile to move. Then c pawn needs to go to 4 or 5 to not block knight, while d remains in place blocking bishop and queen. Now if we have e remaining in place too, even Q has a single c2 move. On the king side, everything is mirrored except g can go to g4/5 as well.
So, here we have partial solution (white pieces are capitalized, black are lowercase, O stands for possible position):
This requires at least
6-9 moves from white (so, 10-7 from black's pawns). If you select 8 moves each:
The following position is reached
26 pieces with a single move - all 16 white, 8 black pawns and 2 black rooks.
Obviously, to improve the problem is clear:
to squeeze black pieces, a lot of moves will be needed, so this probably won't be the optimal solution in terms of number of chess moves
One possible end solution is:
6 white moves, and 10 moves from black pawns + 2 moves from bishops, 3 moves from king, 1 move from rook and 2 moves to get queen in place.
This is in total
24 moves - 2 better than the current solution with 13 each; however, it takes 18 turns in chess due to so many moves needed from black.
As for a proof game:
1.a3,a5 2.b3,Ra6 3.c4,d5 4.f4,d4 5.g3,e5 6.h3,e4 (from now on, odd white moves are Ra2, even are Ra1. Only black moves are shown from here on). 7.Ra6 8.Qd5 9.Qa8 10.Bb7 11.c6 12.f6 13.Kf7 14.Kg6 15.h5 16.Kh6 17.g6 18.Bg7