Welcome!
Have you checked fuse F1 on the rectifier board? It's the feature lamp fuse that's inline (upstream) from the BR1 bridge.
It's a 10 amp fast blow fuse. Take the fuse out of the rectifier board and test continuity across it.

Paragon Manual & Schematics are downloadable from the Internet Pinball Data Base (ipdb.org) - see here:
https://www.ipdb.org/machine.cgi?id=1755

Pinsider @inkochnito has also created lots of useful tech charts - grab the Paragon one from the Bally side link:
http://www.inkochnito.nl/
BTW @inkochnito if you're watching, you're missing the 3 bank drop target reset coil details from the Paragon coil list

Schematics indicate Q17 is for the 3 bank drop target reset coil.

Ok, on that coil there is a diode soldered across the lugs. Check that you soldered the yellow wires to that coil where the banded side of the diode is connected to. If the yellow wires are on the wrong lug, that diode on the coil will now be blown and have to be replaced.

Next, leave J5 disconnected from the solenoid driver board, power on and measure the voltages at the following locations on the solenoid driver board:
Pin 10 of U4
Pin 9 of U4
The banded side of diode CR17 (just above Q17)

These voltage readings will tell us if a failure in the pre-driver circuit is causing the Q17 driver transistor to lock on at power up.

Sure, the back multi-meter lead needs to connect to ground. The solenoid driver board has a "GND" test point which is ground. The backbox also has a ground braid wire running along the bottom and I just clip the black meter lead under it because it's more convenient.

The voltages you're measuring will be between zero and 5 volts. If your multi-meter isn't auto-ranging set it to 20 volts DC range.

The red multi-meter lead needs to touch the measurement points I mentioned in the previous post.

From memory:
Pin 10 of U4 will read about 0.8 - 0.9 volts
Pin 9 of U4 will read about 0.3 volts
The banded side of diode CR17 will read about zero volts

Click on the image below to zoom in to find the measurement locations.

Rectifier_Board_Q17a.jpg

Looks like that pin 10 of U4 has failed and internally gone open circuit. You'll need to replace that CA3081 transistor array chip at U4.
For your sanity, you can measure the voltages at pins 10 and 9 of U1 and U3 (they're both CA3081 chips too) to see what you should be getting.
The diode at CR17 is a standard 1N4004. It's probably ok.

With the relay installed - don't soak the relay by washing under running water or in dishwasher.
If dust and just plain dirt, can carefully run underside and part of topside under running water. Avoid water coming in contact with relay.
For cleaning flux - flux cleaner works best but alcohol also does the trick.

Main filter cap is from 1981 - time for him to be replaced.

And read this:
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern

Pay particular attention to this page:
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#AS-2518-147_ground_updates_.2F_bullet_proofing
My board had pin 10 of J3 burned which wouldn't have happened if these updates had been performed.

Check connectors for dark or burned areas.
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#Connectors
And look at soldered pins for cracks around pins, reflow if necessary.

Little correction, that's for a different (combo lamp and solenoid) driver board which is only in a couple of later games.

You want to look at this link for your solenoid driver board :
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#Solenoid_Driver_Upgrades

Original was 11700uF @ 20V. Odd size and not real common.
Easiest to find replacement size would be 12000uF @ 25V.

Original spec for that top dark blue capacitor at C26 is 160uF @ 350V.
Ed sells an electrolytic capacitor kit for that board listing his part numbers for each capacitor:
https://www.greatplainselectronics.com/proddetail.asp?prod=BALLY-P22E-KIT

If you are just talking dust, use a dry paint brush to brush it away.

Measure the voltage (red meter lead) at test point TP3 on the solenoid driver board (black meter lead on ground). It should read 5 volts DC. If it doesn't you have a bad connection on your J3 connector which has a loop wire between pin 13 and pin 25 that connects 5 volts to TP3 (and then to the solenoid driver circuit on the board).

One of the bullet proofing mods G-P-E alluded to on Pinwiki was to hardwire 5 volts to TP3.

Yes, U4 is oriented correctly.
See my previous post.

What about TP3 on the "solenoid driver board"?

The length of some tones played by the sound board can be sustained shorter or longer to blend into adjacent notes played, however from memory on your particular game it doesn't have much useful effect because it was in part disabled - that adjustment is more useful on the earlier games that create slightly more complex sound effects.

Great to hear, playball!!

You mean G.I is working in the back box, but not on the playfield? That can only be a connector issue at the rectifier board. If you wiggle connector J1 on the rectifier board do the lights come on?
Fuse F5 on the rectifier board is responsible for G.I but if backbox GI is working then it's not the fuse.
The lamp driver board is only for feature lamps - it plays no part in G.I.

Hmm, there are two wires for each G.I power line to the playfield, rectifier board J1 pins 1 and 2 (white wire and green wire respectively) are ground, pins 5 and 8 (orange wire and red wire respectively) are power.
Would seem unusual that two wires go out. Are you sure the G.I in the backbox is ok?

Do you measure around 7.3 volts AC at test point TP4 on the rectifier board? (you need to set your multi-meter to measure AC voltage). Is fuse F5 on the rectifier board (20 amp fast blow) still testing good?

With your meter still set on AC voltage, what do you measure directly across any G.I lamp socket under the playfield?

Did the G.I suddenly go out or did it happen while you were working on something?

Yes, pull that J1 connector off the rectifier board.
The G.I. in this game is AC so always set your meter to AC when diagnosing G.I.
Black meter lead on pin 1 of that J1 connector, you should measure 6.7VAC at pin 5 and pin 8.
Do the same again but the black meter lead on J1 pin 2.

Where was your black meter lead when you were taking these measurements?

Yes, J1 is the power connector for the playfield that carries G.I, feature lamp power and solenoid power.

Your rectifier board shows thermal stress on the copper trace at pin 8 of J1 - maybe the trace has burnt open there.
Just solder a thick wire on the back of the rectifier board at the trace running from the top of fuse F5 (find an convenient spot to solder to on that thick trace) directly to the trace connecting to pin 5 and pin 8 of J1 pin header. Note your jumper wires should be equal or greater in copper diameter than the two wires coming off pin 5 and pin 8. They are carrying heavy current.

Also verify there is good soldering on the GI ground connections of that J1 pin header at pins 1 and 2.

If you decide to replace the rectifier board go for the one here that's produced by pinsider @barakandl:
https://nvram.weebly.com/new-pcbs.html

Sounds like a bridge rectifier may be toasted if your feature lights are out now - check those - that one on the right looks like it has gotten pretty hot -

At this point, for $55 shipped, there's no shame in getting a brand new replacement. If you are decent at soldering, you can get the kit unassembled for $35 shipped.
That board looks like it's been stressed, stitched, jumpered, and hobbled along for years now - let it die a peaceful death.....just my 2 cents worth....

https://nvram.weebly.com/new-pcbs.html

Try a different bulb - - - -

You can search the forum here on how to test the rectifiers - I'm actually punching out of work right now and hitting the road -

Yes, J1 pin 1 and 2 should connect to ground - the trace is on the top side of the board. It looks like the solder blob on those pins hasn't connected to the copper pad under the solder from the copper being tarnished. Remove the solder and clean (scrape) the two copper pads till you get a shiny copper color and resolder.

Pins 1, 2, 3, 4 on the J3 connector below it are also ground. So for redundancy solder a thick wire from there to the two pins at J1 pins 1 and 2.

Take a few photos showing all the connections before you start removing wires so you have a reference. The wires connect to points that are "E" numbered, i.e. the front picture board you posted above shows locations marked E1 through to E12 - that's where the transformer wires connect to.

There's a guide on PinWiki:
https://www.pinwiki.com/wiki/index.php?title=Bally/Stern#How_To_Hook_Up_a_Bally_AS-2518-18_Rectifier_Board

Go with whatever method you find easier, just triple check all the wires are at the correct locations. Feel free to post pictures when you're done if you want another set of eyes on it.

Good luck!

All looks correct to me.
Connect J2 only.
Power up. If there's no smoke (just kidding), measure the test points on the rectifier board - the voltages are listed in the schematic and on PinWiki.
Make sure to set your meter voltage range accordingly before measuring each point.
The G.I is AC, all other voltages are DC.
Black meter lead goes on any ground point.
If the voltages are in spec, then hook up J1 and J3.

Good luck!