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 )
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| 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 |
