The parity argument is a well known argument telling you that a Knight takes an odd number of moves to go from b1 to h8, no matter which route is used, because b1 and h8 are squares with different colors. Similarly, a Knight takes an even number of moves to go from a1 to h8.
But the parity argument applies to other situations (Knight moves are the simplest case). Consider the following schematic problem:
15+16. Who has the move?
Here each Rook did play an odd number of move. The white Knights did play an odd total number of moves, and similarly for the black Knights. The white Queen has been captured at home (by a bN) and the white King did play an odd number of moves. There remains to account for the 4 moves played by the a and h Pawns.
Summing all, we conclude that an odd total number of moves has been played, so that Black must have the move. Indeed, the most common use of the parity argument is in "Who moves?" problems.
Problems using the parity argument are also called to display the "odd/even theme". Here is an other application:
T. R. Dawson
The Chess Amateur, Feb. 1927
15+16. Black to play. Indicate a move Black must have played