I've lived in a circa 1810 house with an unmortared stone foundation for the past 32 years, so...

Stone foundations and bone dry cellars are probably not possible. In many cases these cellars incorporated a gravity drain to allow water to exit. These can clog up over time, so I'd certainly check to see if there is one. They are almost always located in a corner and there is usually a system of perimeter gutters in the cement floor to guide the water to the exit point. Water has to go somewhere and early stone foundations generally allowed it to enter and leave without doing any damage.

An interior drain system may work (but I'd bet against it) and I would not want to try drilling into the foundation itself.

Grading to direct water away from the cellar walls can do a great deal of good and does not have to be fantastically expensive. A simple and effective system is to attach heavy duty poly plastic sheeting to the foundation and slope it away from the house. This requires some careful exacavation and probably some fill on top of the plastic, but it works well. No doubt you can find information about this system on line. Extending downspouts so water runs away from the house will also help.

I assume the parging was done on the interior walls. I think I would want to know if this was done for purely cosmetic reasons or to hide some more serious defects.(Serious cracks if this is mortared stone, for example.) I had my own interior walls reinforced with wire mesh and sprayed on shotcrete 3 years ago, so I don't mean there's something automatically sinister about the parging.

If there had been heavy rain just prior to seeing the standing water, perhaps it's not a common occurence at all. Adding a sump pump for such rare occurrences is a possibilty. I would check the cellar carefully for evidence of rot in the sills, joists, etc. If they are sound, chances are excellent that water is not an issue.

There's no reason to reject a house you like because of a stone foundation. You're probably not going to be turning the cellar into a family room, but that's just part of living in an old house. Stone foundations can and do last for 100's of years - my neighbor's was built in 1778.

Did the waterproofing co. do any analysis of where the water is coming from?

The first order of business is to figure that out. The simplest solution may be gutters, downspouts, and drains to carry the water away from the house. However, if the house is located on low ground or, say, in an underground water channel, then that won't help you much.

You might ask the current owners about moisture problems and what they have done.