I'm confdient Bartolini will get back to you with their explanation soon, but meanwhile let me have a go (and let's see if the two match
That's similar what I had under the pickup I checked with them. IIRC the original quad-coils used to have an XX designation at the end of the model number, but just like mine - yours looks to have the 'lugs' of a quad-coil, with a dual-coil label.
Looking at your top picutre, label the outermost 'lugs' as follows
A B
Z X
In your pic, B is grounded to the shell. Break this connection & B becomes your Green (on bartolini diagrams).
Cut the red wire that's currently connecting Z-X.
X becomes 'Red', and Z becomes 'white'.
Finally you have a wire that is black with an inner 'transparent' sheath, going to A, and the sheath is grounded it to the pickup body. A is your 'Black', and shield is... well, attach it anywhere to the copper plate.
This (if i got it right) gives you the standard Bartolini 4-color double-coil wiring. IIRC Red+Black to hot, Green+White to ground should put it in parallel? (double-check with the Bartolini charts, just to be sure).
But, as you noted: you also have another set of lugs on the pickup:
A B
a b
z x
Z X
With a-z bridged, and b-x likewise bridged. Breaking this bridging would allow you to turn this into a full quad-coil, which of course might lead to a new variety of wiring options.
Good luck, and let us know how it goes
<Second thought: if you're only wanting it in parallel and don't want the various coil switching options, your 'green' is already grounded to the shell, so just ground Z/White to the shell too & take the X/Red from lug-X across the understide of the pickup to lug-A. Not sure whether the red piece of wire that's there is long enough to bridge X-A, but it might be? This would give you perhaps the least-hassle way of going from series to parallel, and you'd need to do a lot of re/de-soldering >