Yeah, it [as in Fermat's proof] probably doesn't exist. His margin comment sounds too much like an excited graduate student!
It's not really a "new form of subtraction". Every form of subtraction, ever, works with the principle of trying to take single digits away from single digits when it gives a non-negative number, and adding and subtracting ten if it does not. The only difference between the methods is where you put the ten, and how many intermediate scribblings you allow along the way.
In that sense, it's frustrating to hear that so many parents are left flummoxed by modern methods. I suspect that this means such parents didn't really understand the one they were taught either, although they may have been able to use it all the same. But the point is that once you know one method, you can in principle deduce how any other method of subtraction works.