Getting that parallel dominant mix of S and P is a tricky one. I had a sketch this morning which I thought would do it, but I needed the central position on the 4PDT on-on-ons to have three connected to the left and one to the right, instead of two of each.
I had another go at the parallel dominant version of this switching, respecting the limitations of a 4PDT on-on-on. Here is my attempt, which is half baked and therefore only for intelectual interest. It's similar to semi-solids one, but not identical.
I left off the phase switches and pots etc.
I found that I could not do this without a couple of features that we try to avoid if possible, being coil shunting, and an undesirable all-off condition.
On an SSS, I'm personally not too worried about shunting coils if necessary, since from testing , I found the effects are somewhere between 'negligible and 'not a problem'.
I was not too pleased to find that I needed an open-circuit all-off setting however, rather tha the inherently quieter short-circuit one. This could perhaps be acceptable with a very good internal shielding job. And also I don't play much with all the pickups off. I find that having at least one switched on is much better!
However, I would have to say that I do not plan on building one of these. The switches cost more than most of my guitars, and I find three-position toggles to be fiddly to set, and less positive and clear than two-position ones.