Unfortunately, there's not an easy way to spot or correct parity errors, but understanding what causes them can help you memorize how to correct them, and figure out where to go.
Mechanical puzzles have something called "virtual states." These states are a part of the cube that exists, but they're either covered up or not shown on the surface of the cube. There are a countless number of virtual states, but the importantly: some virtual states, even though you cannot see them, make the difference in solvability. Parity errors most commonly (though not exclusively) result from errors in states you can't actually see.
A common example of what I mean by this is the centers on the 4x4. You can't see them - not the stationary ones that exist on a 3x3 - but they still exist. Parities emerge on the 4x4 when the invisible 3x3 centers on the 4x4 are rotated into an invalid position. You can't see them, but they're there.
As a result, they're quite literally impossible to spot until you run face-first into an unsolvable state. To correct them, you need to execute an algorithm that puts the internal hidden states into their correct locations, likely at the cost of re-scrambling a portion of the rest of the cube.
As far as algorithms go, there's nothing to do but memorize, unfortunately. Trying to understand how the algorithm does what it does will let you remember it more clearly, but at the end of the day, it's still memorization. Practicing them over and over until you have them down is, at some point, the only way.
Here are some of the shorter algorithms I've seen that fix parity errors:
- 4x4, when one edge is flipped on top front:
(Rr)2 B2 U2 (Ll) U2 (Rr)' U2 (Rr) U2 F2 (Rr) F2 (Ll)' B2 (Rr)2
- 4x4, when two edges need to be swapped; one in front, one in back:
(Uu)2 (Ll)2 U2 l2 U2 (Ll)2 (Uu)2
- 4x4, when two edges need to be swapped; one in front, one on the left:
L2 D (Ff)2 (Ll)2 F2 l2 F2 (Ll)2 (Ff)2 D' L2
- 4x4, when two adjacent corners need to be swapped:
(Uu)2 (Ll)2 U2 l2 U2 (Ll)2 (Uu)2 F' U' F U F R' F2 U F U F' U' F R
- 4x4, when two opposite corners need to be swapped:
(Uu)2 (Ll)2 U2 l2 U2 (Ll)2 (Uu)2 R U' L U2 R' U R L' U' L U2 R' U L' U
(4x4 algorithms pulled from this page.)
Fixing 5x5 parities are a little easier:
- 5x5, when you have two or four edges to swap remaining, and swapping them is not nominally possible:
- 5x5, when one edge has both its wings flipped upside down:
(Rr)2 B2 U2 Ll U2 (Rr)' U2 Rr U2 F2 Rr F2 (Ll)' B2 (Rr)2
- 5x5, when two wings on opposite sides of the cube need to be swapped:
[(Ll)' U2]2 F2 (Ll)' F2 Rr U2 (Rr)' U2 (Ll)2
(5x5 algorithms pulled from this page.)
On higher order cubes, solve from the inside out, and use parity algorithms as you go. For example, on the 7x7, pair the inner edge wings with their edges, then the outer edge wings with their edges. If you run into a parity case on the inner edges, it'll be easier to spot and less destructive to correct if you go inside-out than outside-in. The same algorithms will work, though.