Go into switch test and check for closed switches , Maybe a short on the slingshot switch lines
Check Left and Right bumpers, Swinger target, Saucer and Outhole switches ,
or switches on the other lines to check are the Top left Bumper lane and 'P' Target , But I think it would be one of the other 5 I mentioned that could be the cause
good Luck

It could be that another switch on the Switch Matrix with a bad diode, is triggering the slingshots, especially a switch with a capacitor inline with it.

Like Ralph says, take the balls out, go to Switch Test (little red button on coin door), and make sure the proper number is being displayed when each switch is pressed.

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This is something you kinda just have to live with in bally world. At least to some extent.

There is a really long thread on pinside somewhere about it, but basically the flippers introduce noise to the switch matrix because the wire harness is all bundled together. It is usually happens when you release a held up flipper. The motor that does the swinging target in my black pyramid will make a nearby pop bumper go when it turns off sometimes too.

If it really bugs you, shield or isolate flipper wires. I also read a comment on RGP recently that said if you can live without the 0.047uF cap across the switch it is less likely to pick up noise, but I havent tried that. You could see if the slings have the cap on them, I forget if bally used them there... def on the pop bumps tho. I find stand up targets are the only switches that really need a cap. A properly adjusted pop and sling will work fine without the cap.

Yeah, I see your point; I missed that you said it does not score.

As Andrew says, some Bally do have that phantom coil problem.

Put new diodes on the Flipper coils and anything else that seems to be sending a pulse.

Some latter Bally have the .047uf 50V capacitor on the pop and sling switches (like Vector) and some don't, so you can always try adding or removing it for an experiment.

You can also try and put a 2.2uf 250v spark suppressor cap across the flipper EOS switch like Williams games have. Williams experimented with that on some of the SpaceShuttle and Sorcerers, and then brought it back for games after F14. Might be worth .75 cents to try.

https://www.greatplainselectronics.com/products.asp?cat=34

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This was not an issue when we used to route these games new.

If you ever restore the playfield, put the entire playfield wire harness into the dishwasher and strip off all that conductive carbon. That seems to work almost every time, but it's obviously only worth it if you are removing it anyways.

https://pinside.com/pinball/forum/topic/cleaning-playfield-harnesses-vids-guide

The software has a provision for the phantom reads too. Bally talks about it in the MPU theory of operation booklet which you can find a PDF of. From memory it say the MPU constantly scanning the switch matrix at 120hz. If it sees a switch go closed it takes memory of that it is closed. If that switch is for a reactive solenoid like a pop bumper or a sling shot the computer will also fire the associated solenoid. On the next scan of the switch matrix, if that switch is still closed it only then awards points. Usually the phantom pops / slings do not score which makes it easier to ignore.

Here is that thread you are referring to covering this & all the other possible causes.

https://pinside.com/pinball/forum/topic/pop-bumper-fires-with-flipper-activity/page/5

Before getting too deep into this I reccomend you determine if the problem is electrical or mechanical.
Start by delsoldering the switch wires from the sling shot switches and see if the problem goes away. If it does it's mechanical.
Make sure the sling rubbers are not too tight. Thats the number one mechanical reason for a random popping sling shot.