Try rubbing alcohol or denatured alcohol.

plastic bread closer....it should work the same as your finger nail, just a little bigber and not so hard on the nails!
The fact that it's plastic as opposed to metal means it shouldn't dig into the varnish underneath.

There are many things that will dissolve latex paint & not an oil based type varnish. I am guessing that you might have a knotty pine tongue & groove paneling. Whatever you have if you try to dissolve the paint then its gonna get into all of the cracks & crevices & then be practically impossible to remove.

If I were in your shoes looking to accomplish the same thing here is what I would do......bite the bullet & find a tool that effectively scrapes the paint off the flat areas. This might be a well worn putty knife or 5-1 tool or a carbide paint scraper. Buy a contour tool with multiple blades to scrape out the crevices. Use "Goof Off", lacquer thinner or whatever solvent is working for you, put it on a rag & rub on the missed areas. Use a permanent brown sharpie or whatever is close & color the areas where the paint can't be removed, sand down all of the varnished pine & apply 2 coats of poly or varnish for the best results.

Sounds as if the old finish was not properly cleaned/prepared before applying the latex. A not-smart way to do things, but perhaps a bit of a blessing for you in this instance. Scraping will be necessary, but the alcohol might make the job much easier and faster. See if multiple alcohol applications will help. Be sure to keep the place well ventilated when using the alcohol. Try ammonia on a small spot.

I had tenants that painted my new Kraftmaid cabinets with random strokes of paint (the wife didn't like the glazed pinholes that I paid extra for). I used Goof Off. It worked great. I would gently scrape as much as you can off before the Goof Off. You will need a lot of rags, preferably ones with a little texture. My cut up old towels worked better than the shop rags from HD. Make sure it is well ventilated; I definitely lost some brain cells that week. lol. The directions said it may degloss the finish slightly but mine looked brand new (which they were).

"I used Goof Off."

Goof off is Xylene in small can at a huge markup.
It will remove dried latex paint, but can damage some finishes.

I am having the same issue. I know it was 13 years ago! But did you use goo off and did it work?