from Patrick Johnson:
BHE = back hand english = setting up for a centerball hit then moving the grip hand (back hand) sideways to apply english and compensate for squirt/swerve at the same time. Works best with higher squirt cues where the "pivot point" is somewhere near the bridge. An approximation that must be fine-tuned by feel.
FHE = front hand english = setting up for a centerball hit then moving the bridge hand (front hand) sideways to apply english and compensate for squirt/swerve at the same time. Works best with lower squirt cues where the "pivot point" is substantially behind the bridge. An approximation that must be fine-tuned by feel.
Parallel english = a bad name ("parallel" doesn't really apply) for simply placing both hands as necessary to apply english and compensate for squirt/swerve. Works with any cue, but is done entirely by feel and can be more of a challenge for some players without the initial approximation provided by BHE or FHE.
All of these are different ways of getting your cue to the same position. Only one cue position will produce the exact shot and spin you want.
Backhand english isn't an exact science - where you place your bridge for the pivot changes from shot to shot depending on how much swerve there is in the shot (which depends on shot length, speed, amount of sidespin, elevation of the cue, cloth cleanliness/age, ball cleanliness/age, even humidity). Backhand english can be a useful technique despite all these variables - it will still get you closer to the correct aim adjustment for any shot, but you'll almost invariably have to make some small additional aim adjustment "by feel".
And here's the kicker: it may or may not be a good thing to know about the "by feel" part of backhand english. Most backhand english users believe (or pretend to believe) that it's an exact adjustment for every shot, which allows their subconscious to make the final adjustment without being confused by "too much thinking". Colin Colenso, on the other hand, makes all the adjustments consciously, even mathematically [for more info, see: aim compensation for squirt, swerve, and throw]. You'll have to decide how to do it for yourself.
Personally, I'm somewhere between the two extremes of "totally by feel" and "Colin Colenso total consciousness". I try to think about all the variables as I'm lining up the shot, to be sure my subconscious has all the data it needs, and then I try to let that all go just before I shoot, getting out of the way of my subconscious for the final aim and stroke refinements