Best way to parge exterior stone foundation wall
I have used wire mesh attached to stone to stucco stone walls, I doubt your wall is moving after all these years but the mortar possibly could use some repointing but the stucco layer would take care of it by taking the weather.

Added to answer OP questions. The stucco mix is sand, lime,cement and water on larger jobs I pick up a trailer load of sand &amp; bags of lime and Portland cement, for smaller jobs I pick up a couple of bags of pre mix. When Covering stone work I use expanded metal mesh because it requires less fasteners than wire, I have tried to use concrete nails but they don't hold well the reason we doing this in the first place was because the exterior had similar problems like you are having, a powder set driver just blew out chunks of mortar. What I did have very good luck with was tapcons screws, I was able to drill the mortar and if not overtightened they held fine and drilling the pilot holes was quick in the mortar. Where there were large stones at the corners I drilled the stone but that takes much longer than the mortar. We mixed our batches 5 gallons of sand about a gallon of lime, and a small coffee can of Portland cement maybe a quart , mixed by hand in a mud tray maybe 2'×3' about 8" deep. Mixing in small batches is important if just 1 person is applying just enough water so a trowel load when placed in a pile will stand up 4-5" damp but not wet, with the expanded mesh only 2 coats was needed the first worked into the mesh then the top coat. I am not a pro at this because I don't do it all the time but have done complete houses and a bunch of foundations where 1 job lead to 1/2 dozen more in the same neighborhood.