Ron,
I forget the disassembly of a Parker as it's been so long since I did it. I do have one apart that has been apart for years. But there's a tutorial somewhere on this site about disassembly.
I'd recommend you go forward. Little harm can be done to this gun.
Ring the barrels to see if they've delaminated.
Closely examine the rib solder line for failure with magnification and good light
(looks like you may have some solder "touch up" to do at the muzzle as a minimum)
Examine the bore and chambers for pitting that would make the gun unsafe (damn few get this bad)
Check the barrels for being on face without the forend on.
Fix any of the above problems before proceeding (espcially off face problems).
Now for the fun...
Have the frame annealed first
reassemble the toplever and locking mechanism.
assemble the barrels on the action before beginning filing of the barrels.
take detailed pics of the engraving (not a biggie here with a Trojan)
draw file all the pitting out of the barrel outside, checking wall thickness as you go.
assemble the forend iron on the barrels/frame before the next part
Draw file all the frame/forend iron pits
polish as an assembly.
If you look at these guns, all the flats are flat and corners are supposed to be fairly sharp. Refer to pics on the web of good original examples. Look at the polishing lines for direction of polishing and grits to use. Barrels typically are taken to a 400 grit (wet/dry paper) finish, same with the frames. KEEP THE CORNERS SHARP (as they were originally). The idea of assembling the barrels and forend iron for filing and polishing is to keep the corners from rolling during polishing which ends up looking amateurish.
I'm sure I forgot some things but that should get the thread rolling on what needs to be done
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