It occurred to me that on the 4x4x4, doing the PLL parity before you do PLL can influence which PLL you will get.
Understanding that OP's idea is to use PLL parity (i.e. r2 U2 r2 Uw2 r2 u2) to influence (improve) PLL. For that to work, the corner permutation AND (d)edge permutation need to be made or kept the same (odd/odd, or even/even). If their permutations are different (odd/even or even/odd), before doing the parity alg, then doing it
CAN be beneficial. On the other hand, it actually makes matters worse - to recognize and change the (d)edge permutation, if the corner permutation is already matching. Doing PLL parity when it is not needed will require another PLL parity after the normal LLperm to fix. Mistakes here can become nasty odd corner swaps as leftovers of that wannabee *better* PLL case. Just in case - here is a new alg I found for diagonal corner swap.
reCorner()™ = (U2 Rw2 U2 R U2) x (U2 R2 U2 R' U2) x' (U2 Rw2 U2 (L'M') U2) r2
r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2 being the standard alg that you spam when you see an off OLL and preserves PLL, but has anyone mastered recognition for the parity fix alg?
r2 B2 U2 l (r2) U2 r' U2 r U2 F2 r F2 l' B2 r2 - this is cmowla's (r2) trick to convert OldStandard(OLLparity) to OldStandard(OLL+PLL)parity. Wide slice versions are also possible, so that can be a very useful multi purpose parity alg.
(l2 B2 l U2 l U2 (x') U2 l U2 l' U2 l U2 l2 U2 (x)) solving the OLL parity and fixing PLL parity at the same time (or in the case of incorrect recognition turns a good PLL into parity?)
To fix OLL and PLL (make all corner,edge, and dedge permutations EVEN) at the same time:
reParity™ = (Rw U2 Lw' U2) x' (Rw' U2 Lw U2 Rw' U2 Lw U2 Lw' U2) (S2)
and it's slice version for the other double parity, = (r U2 l' U2) x' (r' U2 l U2 r' U2 l U2 l' U2) (M' U2)
Netting 2 birds with one stone.