Games People Play - Allen Parsons Project
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...sound in vogue by the mid-'80s (but it's close). It might be a DI track in parallel with the amp sound, and the guitar may not actually be a Les Paul--if so, that's either a coil-tapped or a parallel humbucker sound. My best guess is a Tele or a Strat on the #2 setting.
You'd be amazed all the crazy things that are done to get guitar tones in the studio. One of my favorite stories is how George Lynch used one of those crappy little Rockman things to get extra definition from massively overdubbed distortion tracks. He would lay down like 8-12 overdubs of totally saturated tones, and then he would DI a cleaner (but still distorted) Rockman track or two to balance out all the fuzziness from the tube amps. The Rockman sounds pretty lame on its own (diode/LED hard-clipping), but it turned out to be magic in parallel with the tube amps melting down. 
My guess is a Tele with both pickups on.
I do think I could get that sound with my JS-1000 with coils split and both pickups on. But, I'm too lazy to bother. 
I mostly play acoustic (and, of course, piano) because all I have to do is grab a guitar off the wall & play (or sit down at the piano).
No cords, no power, no twisting knobs....
Just play!