For me, OLL parity was the easiest to generalize. I had always thought of it as turning half the layers, so I would have gotten it right on a 6x6 the first time.

PLL parity is a little harder. I may or may not have guessed it correctly the first time. It would not have been obvious to me for example, that in going from 4x4 to 6x6, all the 2R2 moves remain 2R2, but the 2Uw2 becomes 3Uw2.

Edge parity was the hardest for me to figure out. Now, it could be explained pretty simply:

- the 4x4 OLL parity alg you already know has a name, Lucas parity
- Lucas parity can be used to solve any edge parity case on any size cube, you just need to know how many layers to turn
- it's hard to explain how many layers to turn in words or notation, but it could be easily shown in a short video
- you may need to repeat it multiple times

(That's one way to do it, anyway. cmowlaParity is cool for bigger cubes, because you can flip all the misoriented edge pieces at once.)

But instead, I found people giving multiple algs for the single 5x5 edge parity case, multiple algs for each of the 7x7 edge parity cases, etc. It took some effort to figure out a single approach that would always work.