You need to change out some connectors.
On solenoid board - replace plug on J3 and ditch that old IDC type connector.
New plug = Molex 22-01-2257 (obsolete) or CS100-25-LR.
You can find CS100-25-LR, contacts (08-52-0123) and key (15-04-9209) here:
https://www.greatplainselectronics.com/products.asp?cat=164
The wire that he spliced onto belongs at location 10 within the new plug.

Replace entire set of plugs going to rectifier board:
CS156-08-LR
CS156-09-LR
CS156-20-LR
Contacts (08-52-0113 or 08-52-0072) and 3x keys (15-04-0219)
These can be found here:
https://www.greatplainselectronics.com/products.asp?cat=81
They probably soldered it because 20 pin plugs used to be somewhat difficult to find.

You will need to replace the PC board mounted headers on the rectifier board. The existing ones are either burned or damaged due to previous person soldering wires directly to the header pins. New header = 26-48-1241
https://www.greatplainselectronics.com/proddetail.asp?prod=26-48-1241
You would need two of these to replace the existing headers.

Alternative to replacing header pins is to replace rectifier board. Can get new ones but don't spend more than about $70.
See here:
https://nvram.weebly.com/bally---stern-rectifier.html

For good info -- see here:
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern
And especially here:
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#Solenoid_Driver_Upgrades

You can find the color codes within the Dolly Parton manual, sheet 45.
If you don't have the manual - you can either buy one (Pinball Resource?) or download a soft copy from ipdb.org

Yes, the wiring on yours looks pretty buggered. First wire is see on your board is the wire at pin 1 -- white or color "50". Dolly Parton manual says this goes to pin 3. But pin 3 already has a gray/blue (92) wire installed... No doubt the opposite end of these wires are going to the wrong places as well.
Incidentally, Pin 1 of J3... I cannot find anything in the schematics of the solenoid driver board that uses pin 1.

Only easy wire to spot is the missing (damaged) pin 10 should be the white/brown wire that is jumpered to the capacitor lead on your board.

No easy fix, you will need to go through all of your wiring *at both ends* and untangle this mess.