Hello all. Huge guitar nerd and massive PJ/JP fan, as well as her bandmates' other work.
Some of the unknowns pictured above are: Gretsch Jet Firebird (Polly pic), a couple of Hagstrom IIs (Josh Klinghofer, one white and one black). The Danelectro U2 JK is pictured with is a Baritone- the neck is way longer than a regular U2. Not sure about the hollowbody he's pictured with- the photo is too small/blurry and 404s when clicked on now. Looks like possibly a vintage Hofner though.
John Parish's black Telecaster is a '68 and likely all stock aside from the "Andy Summers" style bridge. It is still in JP's possession and was his first electric guitar, and is as it was when he got it plus a peace symbol sticker. This info is from Mr. Parish himself.

I recall somewhere a photo AWAMWB era of JP's pedalboard, with an Ibanez AD80 delay, maybe an MXR Distortion plus, and a couple of Joe Gore-built pedals, a Boring boost and another unidentified JG build. Also JP had switches for a signal path with and without a Copicat IC300 tape echo that can be clearly seen in a video shot during lockdown 2020 on Le Cargo. Also a great song! When I saw JP on tour for How Animals Move in 2003, he had an MXR Distortion Plus, a Diaz Tremodillo tremolo...and that was it. He might have had the Copicat back by the AC30. Sounded amazing. Adrian Utley also played guitar and the whole shebang still stands as probably the best show I've ever attended.
Anyway...I've been chasing some of these sounds and bits of kit for years and occasionally bugging JP and Joe Gore with questions they've graciously answered but any and all of them pull amazing sounds out of literally anything, and I've seen them use multiple bits to do the same parts enough to know it's mostly the player, not the gear.