About 2 years ago, I deciphered the common parity algorithms and made "derivation videos" for them on YouTube.
I started a thread on speedsolving.com around that time. For the links to the derivation videos I am talking about, see the first post.
This is the thread Methods for Forming 2-Cycle Odd Parity Algorithms for Big Cubes
I have packed a lot of original information in that thread (I didn't learn any content within from anywhere, and no one before me figured any of this out), so if you all have any questions, ask away. I probably should start making more videos, but I've been working on a huge Rubik's cube project lately that is eating up all of my spare time.
I have dumped all of the algorithms I have found for the 4x4x4 on this wiki page (my name is Christopher Mowla...I recently added authors names on that page, so every algorithm with my name to the right of it I have found without a computer solver...well there are a few exceptions but not many):4x4x4 Parity Algorithms
Also, if anyone is curious, I believe out of all of the parity algorithms I have found, the following double parity algorithm, being just a conjugate, is probably the easiest OLL parity algorithm to understand, as it doesn't require any knowledge of commutators. It only requires that you know that all OLL parity algorithms have an odd number of inner layer quarter turns. That's about it. It's very straight forward.(B Lw2 U' L' U r u2 b' r2 Bw2 E) r' (E' Bw2 r2 b u2 r' U' L U Lw2 B')
After the moves (B Lw2 U' L' U r u2 b' r2 Bw2 E) are executed, study the pattern of the center pieces in the inner layer right slice
: only red and orange center pieces are in that slice AND they are in the formation that, if you do an inner layer quarter turn, they stay exactly the same. This way we can do whatever we want to that slice and, once we undo the moves (B Lw2 U' L' U r u2 b' r2 Bw2 E) with (E' Bw2 r2 b u2 r' U' L U Lw2 B'), then the centers will not be affected at all (well, on a picture cube they will be, but not on a 6 colored cube).
And, of course, the 4 edge pieces in the inner layer right slice are in the correct formation with respect to each other so that, once the inner layer quarter turn r' is executed, then they swap with each other to preserve the edges. To understand why this works, it's just a matter of understanding precisely which 4-cycle double parity is and then placing the edges in the correct locations (with respect to each other).
NOTE: It doesn't matter how many moves it takes you to create the same setup as (B Lw2 U' L' U r u2 b' r2 Bw2 E), only that you understand that setup. If you understand that setup, you're set. I just tried several paths to find the shortest sequence of moves to create that setup, so don't be hard on yourself or be the least bit intimidated by that move sequence.
For all other parity algorithm types (see the different categories I have made on the wiki page I linked to), they require more knowledge than the double parity conjugate above. If you want to create the single edge flip case from that, you just need to do PLL parity which isr2 U2 r2 Uw2 r2 u2
= r2 U2 r2 U2 r2 + r2 u2 r2 u2
or, better yet, the conjugate (r2 F2 U2) r2 (U2 F2 r2).
I can understand why it is so hard to understand parity algorithms, but, at the same time, most short existing algorithms I have either made myself or have studied and understood to a level that I can see most of them share a similar "tactic". I explain quite a bit about various parity algorithms in the thread I linked to earlier.
However, if you can understand the conjugate I have show above, then you can truly say that you understand how one edge parity algorithm works without any doubt or pretending.