IMO this is no scramble with parity: for me parity is, when you have an odd-number of edges and an odd number of corners (not counting the buffer). In OP you will have an odd number off perms for edges and for corners (and therefore after edges the two corners swapped which you'll have to fix somehow).

An easy scramble with parity is: U

Perhaps these thoughts are helpful:

Sometimes I solve blind with J-perm only (kind of OP light), UF<->UR and URF<->UBR. If I have an odd number of edges solved, the two corners must be swapped and I have to fix parity:

1. I do another J-perm that corrects the corners and they are there, where they have been after the scramble (so you could memo them in place). I also swap the edges which then are not in their right position anymore, but as I will have an odd number of J-perms to solve the corners, they will be swapped back in their right position at the end again.

2. I have memorized the corners swapped and "pre-looked" the fact, that they will be swapped after solving the edges. Then I will have an even number of J-perms for the corners and therefore no problem with swapped edges.

A parity-scramble will give you an odd-number of J-perms in total: in the first version you have odd perms for the edges, one for parity and odd perms for corners (odd+1+odd=odd). In the second version you have an odd number for edges and an even number for corners (odd+even=odd).

With J-perms for corners and M2 for edges I would do version 1: just another J-perm after solving the corners (to fix the edges), then do M2 for edges. This will need an odd number of M2 so I will add one in the end and will have the edge-buffer and edge-target swapped. Then I do a "parity-alg" that will swap the edge-buffer and edge target and the two corners.

When do I have parity?

If the scramble has a summ of turns with 90° or 270°! (like a single U)

And look at a J-perm R U2 R' U' R U2 L' U R' U' L (13*90°-turns!): it doesn't swap the pieces as I wrote before, it just looks like that in the end. In fact nearly the whole topface is rearranged and the middle piece is rotated ccw (U2+U'+U2*U*U'=U').

For 4bld: don't do OP before solving the centers (because they will be rotated) - and if the scramble has a summ of 90° or 270° for the inner-slices, there will be the wings-parity.