If you have two coordinate systems with the same origin, you can represent a (linear) transformation of coordinates from one to another as a matrix. This matrix has either positive or negative determinant. This sign of the determinant is what gives the transformation its parity. (All this applies to any number of dimensions, not just 3.)
If you compose multiple linear transformations, the matrix of the final transformation is the matrix product of their matrices. And the determinant of the result will be positive if and only if an even number (including 0) of the original matrices have a negative determinant.
So, you can categorize linear transformations using the sign of their determinant, using their parity. Some (like rotations, scaling or sheering) preserve parity when composed with another, others (like reflection) flip it.
Knowing this, it's easy to see that flipping $n$ of coordinates (regardless of the number dimensions) produces a matrix with $-1$ appearing $n$-times on the diagonal, so the transformation has odd parity if and only if $n$ is odd.