Yup, wood is the traditional material I mean. I am pretty sure that if you took a guitar and cut down the neck, say with a spoke shave, so that it was the same size where it met the body as it was at the nut, it would be very liable to break. That's how wood seems to behave in my experience. And short of breaking I think it'd be liable to be less stable for playing.

With a mountain dulcimer of course the fretboard is fixed to the body all the way along which changes things.

I'm pretty sure it would be quite possible to build an instrument to be played like a guitar or mandolin whuch did have a parallel fretboard. If noone's done it so far it'd be worth doing. Whether it would be worth playing is another matter, but it'd need to be built first.