I am just trying to help
Before it is said & done I will become the master of the Fly's Electronics trying to fix the one I am currently working on..
Just something I noticed looking at your 1st Photo
The jumper I will call it is plugged into a different place than the one I am working on, Mine might be wrong too..
I know enough about electronics to work on homes & cars & am limited to guitar workings but the Fly's are wired different than any others I have seen.
Most of this wiring is fairly simple compared to electronics used now days in computers etc.
I am thinking the capacitors may be the problem on these 15+ year old boards.
Anyhow I will attach a photo of the 2007 Parker Fly Deluxe's wiring showing the board..
What I am calling a jumper is a small black housing that jumps two pins on the board near where the red power from the 9volt battery attaches..
The 2nd photo mmmguitar shows as being the same as the one I have here..
Here's a decent photo of the Fly deluxe Board from 2007
If I have too, I will be mailing this part to Larry Fishman to diagnose & repair..
Only wish my ole buddy Mean Marcus was still with us, He'd have that thing fixed in a few minutes..
I note: That I see cold solder joints which are dull & Not Shiny like they should be, Like they where soldered at too low of heat to join the parts as needed to allow current flow throughout the parts.. Just something I notice that I learned from the Master Mean Marcus.. RIP Ole Marcus..