I believe this is false, and that a counterexample occurs in $\mathrm{SL}(3)$ already.
Let $P$ be the block upper triangular subgroup with a $2\times2$ block in the top left corner. If I understand your terminology correctly, it's "opposite" is the block lower triangular group with the same block.
If a Weyl group element were to conjugate one to the other, it would have to map the root corresponding to the (1,2)-entry to that of the (2,1)-entry, and also (1,3) to (3,1). That is impossible for a single element of $S_3$.