With BHE or FHE, there is the added factor of a constantly changing angle striking the cue ball. With parallel shift the angle stays the same. One less variable. Maximum spin contact point is a little easier to judge with parallel shift, just because of the position of our eyes when shooting.
The contact point on the cue ball can be slightly further out with parallel shift than it can be with either BHE or FHE. I have never seen this proven as an advantage, or disproven. I suspect that the biggest deciding factor on how much spin we can apply is the "grip" for lack of a better word between the cue ball and table cloth. Since that doesn't change, I suspect that the other factors offset each other and to a level far closer than we are capable of judging on a table, the amount of spin that can be put on a cue ball by all methods is basically equal. I know of no tests determining this and have done no significant testing myself so it remains a wag or swag.
I have used a parallel shift most extensively, BHE for long enough periods to be familiar with it, years total, and I have experimented with FHE and combinations of any two. Just being silly I have used all three on one shot! FHE, BHE, and Parallel offset. It works but lordy what a cobbled up mess! I shot with front hand english for about three weeks playing daily. It is superior to back hand english in my opinion, less stick angle.
I can say these things as my opinion or as what I have found. I would like to set up the same repeatability as my old test lab and flog these things but in the end I would be just satisfying curiosity as I know that no form of english offers a huge advantage, well, according to what I have found.