From the musty vault of my memory I seem to recall that Parsons found the lyrics on a paper in the typewriter of his Harvard (or was it Yale?)roomate (was it Buchanan?)and provided a melody and helped him finish the song. The tune reminds me of one from the Carter Family but I don't remember which one.
Re Wild Horses, I read somewhere that the Stones sent it to the Burritos for Sneaky Pete to put a steel guitar track on it (which they didn't use [I think]) Parsons heard it, recorded a Burritos version and put it out before the Stones did. Any corrections appreciated.