I work with a similar situation to what you describe.
When I get complaints (and it isn't only church people, we hire rooms out), I tell the residents to complain to the council as well as me, and I phone the parking services and ask the council to come down and give fines. It is totally unfair on the residents who pay a lot for their parking permits.
I send out reminders and warnings to groups that do this, and always tell new groups not to park there. I make it very clear that they will be risking a fine if they park there.
If someone gets a fine through parking in a resident's spot, then they pay the fine.
And if you came in when I was there I would perfectly happily go into a group and tell people to move their car. I've done it before now, and no doubt will do it again.
At one point it was really bad in the evenings, and I was discussing with the council about them doing evening checks, but we managed to bring it down, and now it's rarely an issue.
You really do need to get the council to come and fine the double yellows and bad parking though, as it is the only way some people learn. I know someone who got four fines before they eventually started parking in a proper place. Every now and then they try it again, get another fine and stop again. If they'd parked in the large multi-storey five minute walk every time they'd have paid a fraction of what they'd paid in fines.
The problem also is if no one is checking them people tell each other "oh, yes, you can park there, I always do." After covid we had a number of people who'd got very slack on where they parked, and they told their friends, who weren't even using us. So the council had to get heavy on them. I think my favourite appeal from that was from one person who appealed the fine because "we always park there and didn't know it was residents only." They included a photo of their car to show, what, I'm not sure. However me and the council had a bit of a chuckle because they'd parked next to the "resident permits only" sign. I suggested the council should fine them double as they'd admitted they'd done it before. Unfortunately (as far as I was concerned-they weren't even using our building) they said they couldn't.
Get the local police/traffic wardens involved.