A parity is whether a move or sequence of moves swaps pieces an even or odd amount of times. Parity can occur if you put the wrong centerpiece in the wrong slot. The role of parity is just to put the correct centerpieces in to it’s right slot and solve the edges in which you got parity
On every face on a 4x4 you have 4 center pieces that each have a specific slot they must go in. You cannot get parity on a 2x2 or 3x3 cube because it is designed like that.
And I didn't mean that if 3x3 is designed in some other way it will get parity. 3x3 means the cube has 9 pieces on all 6 sides, 1 of the pieces on all sides are center-pieces. So, 3x3 has 1 center while 2x2 has 0 and because the 3x3 has 1 center, the center has no other center spots to go to and create a parity, while on 2x2, there are no centers at all!
PLL parity Pictures: (OLL has only 1 parity which is down in the algorithms list)
Algorithm for OLL parity:
Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'
Algorithm for PLL parity:
2r2 U2 2r2 Uw2 2r2 Uw2
What is 2r2?
2r (a big cube notation) means to turn the R move with the nearest opposite side piece to it. 2r2 means to turn that layer twice.
Ok you can get the PLL parity named Z (Given in the parity pictures list) So You can do this algorithm to solve it.
R U R’ U’ 2r2 U2 2r2 Uw2 2r2 Uw2 U R U’ R’
There are some blind memorization tricks to know the odd and even pieces. Now you can do this in 3 situations: if you are doing the scramble or by looking at the scramble or by reading the scramble, but in competitions they don’t let you see the scramble.centre slices, but they are very very hard to cram in 15 seconds of inspection. You can do many other things with that time.
In simple words, you can see the scramble and identify if there’s parity, but in official competitions they don’t allow you to see the scramble.
There are some memorization techniques which can be used to identify odd and evenness of the centre slices. But it’s very hard to memorize in 15 seconds. And 4x4 has many more pieces to solve: 24 centers, 8 corners, 24 edges, in total 56. So, if the 4x4 could be done equally efficiently as a 3x3, it would only take 10.1 seconds to solve it. It's now about 18 seconds, so one could also say that the 4x4 is 1.8 times harder.
I don’t like OLL parity as I am sure I will mess it up almost always.
PLL parity is a bit better for me because it’s a same algorithm with just some moves added at start and end.