No mathematician, but from my experience, this is how I think of it:
1) It seems from prior work on the 3x3x3, that it is not possible to perform a simple exchange of 2 pieces without changing anything else.
2) This means that the simplest moves we can think of do 2 exchanges (meaning 2 edges exchanged and 2 corners, or 2 sets of edges, or 2 sets of corners). 2 exchanges is the same as a 3-way rotation. Think of a>b>c as: a/b exchange, then b/c exchange.
3) Now, transferring to the 4x4x4, all parity really can be explained (at least in my opinion) as: It appears we need to make a single exchange.
Thus, what we call 'parity solutions' really do just that - they make a single exchange of pieces - although often by making an odd number of exchanges.
My own personal "easiest way to see what is happening" solution method goes like this (NOTE - This is NOT fast on the clock):
A) Put centers in place. This is intuitive, and with a little practice goes pretty fast.
B) Place top edges. This is done by pairing the edges as you put them in place, and you have to take care to keep the horizontal center slices correct as you go. It's not difficult to see how as you work with it.
B) Place all top corners correctly. Also intuitive. (This is also the way I first solved the 3x3x3). This step is actually done the same as the 3x3x3.
C) Place middle layer edges (all 8). (There is basically one algorithm I need for this, with a reflection. And, it is really the same algorithm as is used in the 3x3x3 solution, with a slight modification to the 4x4x4 which is easy to see as you watch the move being performed.) The second slice pieces require exactly the same series of moves as with the 3x3x3, and the third is very similar, except that you have to grab an extra slice at one point, and then put it back later.
D) Place bottom corners correctly in place, and then orient. (Again, one algorithm, with a single conjugation for placing. 2 algorithms for rotation. And, these are the same algorithms as the 3x3x3.)
E) Place bottom edges correctly. Obviously, there are 8. I have not paired them yet, so I do this one by one. I do 2 opposite edges first. I need one algorithm with a reflection. It is an algorithm I made up myself, and it does a 3-way exchange. This is really the only NEW algorithm I need compared to the 3x3x3. And, sometimes it needs a quick conjugation.
EE) And, this is the step where 'parity' might be involved. Somewhere in this step I count how many exchanges of pieces I have to make. I want an answer of "even" or "odd." If the answer if "even", then I simply continue until I have all the edges in place.
EEE) If the answer is "odd", I have what is called a "parity issue". Since I try to work the edges by doing pairs together, and non-adjacent sides first, the last 3 or 4 pieces are across the face from each other. This leaves me a simple solution available for the 'parity issue': Since the offending edges are across from each other, I turn one inside layer 180*, then the face I am working with 180* (this effectively puts all 4 edges that I need to fix into the same slice), then the inside layer where all the edges are located 90* either way, then the face I am working with 180*, and finally the inside layer 180* again. So, if the face were on top, the algorithm might look like this: r2,U2,r,U2,r2. small r means the inside layer WITHOUT the right face. Think about what that move does: It does a 4-way cycle on the edges, and also 2 4-way cycles on the centers (which you can't notice, and I am not worried about yet). Thus, I am making an odd number of exchanges (9) and this solves the 'parity' issue.
EEEE) I now continue to put the edges in place.
F) Finally, if I had to do step EEE, then I need to go back and replace the 4 sets of 2 paired centers that were moved in that step.
Admittedly, again, this is not FAST. But I think it makes a great beginners method because most of it is exactly like the 3x3x3.
And, this lines up with what someone said further up: The parity issue is caused by how many interior layer moves you do, because those moves are the ones that do odd numbers of exchanges.
EDIT TO ADD THE FOLLOWING:
Perhaps the truest answer is this: on a 4x4x4 cube, when you rotate an interior layer, you actually accomplish an odd number of exchanges of pieces: 3 for the edge pieces (it is really a 4-way cycle, but that is 3 exchanges), 3 for one group of centers (again a 4-way cycle), and 3 for another group of centers (again a 4-way cycle). Since there is a simple move which accomplishes an odd number of exchanges on this puzzle, then all configurations with an odd number of exchanges when compared to 'solved' are now possible. Thus, what are called OLL and PLL parity are legal configurations.
And, that is why the way to 'fix' parity issues on this puzzle comes down to simply doing one more odd turn of an interior layer.
For those who think in terms of 'hidden pieces,' it is possible to imagine the 2x2x2 block hidden inside of the 4x4x4 cube as "hidden pieces." If you choose to think that way, then the single turn of an interior layer actually does an even number of exchanges: 4 sets of 4-cycles. First, the 3 that I have been discussing, and then 4th, the 4-way cycle of the interior block pieces. As I say, I am not a mathematician by trade, but to me, this is an ok way of thinking about it. However, the group theory of the 4x4x4 cube does not 'require' this idea. The group theory of the 4x4x4 cube is simply different from that of the 3x3x3, because the interior slice move does an odd number of exchanges.
EDIT AGAIN: I revised my simple method above because I learned that you can start with the centers, and it goes a little quicker.
EDIT EDIT AGAIN: What is commonly called PLL parity is not parity at all. It is simply 2 pairs of pieces that need to be exchanged. It only looks weird because the fastest way to solve this puzzle is to reduce to 3x3x3.
What is commonly called OLL parity is really just the need for a 90* turn of an interior slice.