Parity is a 4x4 last-layer case that can never happen on a 3x3. You solve a 4x4 by turning it into a big 3x3: first the centers, then pairing up the edges, then solving it like a 3x3. Sometimes the last layer ends up in a position a 3x3 can't have, and no 3x3 algorithm fixes it. One extra algorithm per case does.
OLL Parity
When you make the yellow cross, you find an odd number of edge pairs flipped: one flipped pair alone, or three. A 3x3 always has an even number flipped, so 3x3 algorithms can't fix it. Do the OLL parity algorithm with the flipped pair in front. After it, you'll have a normal 3x3 case. Finish the cross and the rest of the last layer as usual.
PLL Parity
After OLL, the top is all yellow, but two edge pairs are swapped in a way no 3x3 PLL can fix. It comes in two forms: the swapped pairs are opposite each other, or next to each other. Hold the cube to match the diagram, do the algorithm, and finish with any top turn it needs.
The Algorithms
OLL parity 18 moves
Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'
Not "pure": it also rearranges other top-layer pieces, so finish with your normal OLL and PLL afterwards.
Pure alternative (fixes only the flipped pair): 2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2 Source: SpeedCubeDB
Algorithm: J PermPLL parity (opposite) 6 moves
2R2 U2 2R2 Uw2 2R2 Uw2
Leaves the top layer turned: finish with U2.
Algorithm: J PermThere's no animation for 4x4 cases yet. The diagrams show the top layer, yellow on top, with the first three layers solved. To find the moves for your own scrambled 4x4, parity included, use the 4x4 solver. To learn the whole solve by hand, see how to solve a 4x4.
4x4 Notation
Bigger cubes need a few more letters. Rw turns the outer two right layers together (the face and the slice next to it). 2R turns only the second layer from the right. Uw turns the top two layers. x rotates the whole cube, like R. The face letters, primes and 2s work as on the 3x3. See Rubik's Cube notation.
These are established, widely taught algorithms, not the only good ones. Which feels fastest depends on your hands.
The algorithms are as listed by J Perm, with a pure OLL-parity alternative from SpeedCubeDB (each card links to its source). Every one is checked on our 4x4 sticker model: each algorithm keeps the first three layers, has the stated parity, and returns the cube to solved (with the top turn shown, if any). Every parity position is handled by exactly one case. The check works on sticker colors, so center pieces of the same color could be swapped, which you can't see on a normal 4x4. Diagrams are cross-checked against SpeedCubeDB.