4x4 parity occurs on the last layer of a 4x4, where you get a case that is impossible to get on a 3x3 so you need a specific algorithm to solve it. PLL parity specifically occurs because two adjacent edge pieces are swapped diagonally with 2 other adjacent edge pieces. Generally you can't recognize it until you are at the PLL stage of solving.
PLL Parity case 1:
This swaps the 2 edges opposite each other.
Perform the algorithm with Yellow on top.
r2, U2, r2, Uw2, r2, u2
PLL Parity case 2:
This swaps the 2 edges adjacent each other. Perform the algorithm with Yellow on top.
(R' U R U' ) r2, U2, r2, Uw2, r2, u2, (U R' U' R )
|Rw||BOTH Right Layers|
|r||INNER Right Layer|
|Uw||BOTH Top Layers|
|u||INNER Top Layer|
|X||Turn the cube clockwise around the X-Axis|
|X'||Turn the cube anti-clockwise around the X-axis|