I got lost in the first sentence and I didn't understand any other. No offense but I don't think much other 12 year olds would understand that. Thanks for the article though I will definatly read it some other time.
Basically, parity is an "error" in reduction of NxNxN->3x3x3 which represents a set of states where the cube cannot be solved using the subset {R,L,U,D,F,B} (the 3x3 subset). This is essentially becuase of a "mistake" in the first phase of a 2-phase solution (the stages being reduction, solve like a 3x3x3)
To explain this, let's ignore 4x4x4 for now and focus on a similar problem on 3x3x3. Our first phase will be the F2L and the second will be the LL. As we can tell all the pieces apart, there will be no issue with the LL as it will always be in the standard OLL/PLL states.
However, imagine if we could not tell the orientation of one of the corners in the F2L. This means that suddenly there would be a whole series of new OLL positions that could not be solved using standard OLL because all the corners could be oriented differently. To fix this, we can simply add an extra algorithm which twists the F2L corner back to the state it would be in if we could tell the difference.
For Speedsolvers using a reduction method such as Yau or Hoya etc, a similar situation can occur in two ways: Permutation and Orientation (you can't actually get a permutation parity on odd layered cubes but as this is on 4x4x4 on a more basic level so I will not be adressing that).
For orientation parity, this is where there is a "flipped edge". Or an odd number of dedges misoriented or oriented. This is because there are pieces which appear identical to a human but the mechanism of the cube treats differently. In this case, the wings of the cube have been swapped so the algorithm to fix it will swap those two pieces back (there is another reason with centres but that is currently unimportant).
For permutation parity, this is where there is only two edges or corners swapped- a state which is impossible on a regular 3x3x3. This means that they must be swapped again to get to a normal 3x3 state. This is due to centres being in the wrong place rather than edges.
If you want a more in depth explanation, PM me and I will also be able to clarify anything which you didnt understand.