The following table shows the Parity (0=Even, 1=Odd) for each Orbit Type for a Quarter-Twist (Face or Slice) applied to a solved Supercube. Each value is the sum (mod 2) of all cube piece orbit parities for the same Orbit Type. For Face twists, Parity of X-Center and +Center Orbit Types varies by cube size and the value is the result of equation: V = Floor((N-2)/2) mod 2.
The following table shows Cube Parity for Quarter-Twists by Cube Size. Slice twists are not applicable in the fixed-center model for cube size less than three. Obvious patterns emerge.
To see how the Face values are obtained look at the following table which includes the V values. For cubes with size > 3, Parity is always Even for odd-size cubes because the sum of the V values for X-Center and +Center parities is always Even (they are always both Even or both Odd) as is the sum of Corner and Central Edge Parity (they are always both Odd). Therefore, the V value (for X-Centers) only comes into play for even-size cubes and since it is summed with Corner Parity which is always Odd. The result is that Face Quarter-Twist Parity is the toggled value of the V value (for even-size cubes).