No, I don't know William Elliott's parents' first names. But Dr. John Mitcheltree is listed as his uncle, so you are right his mother is a Mitcheltree.
Furthermore, recent research I've done has unearthed a court case involving Dr. John Mitcheltree's estate. And it seems from those documents that there were eight Mitcheltree siblings. I have found multiple family trees that list the doctor as having one unknown female sister, and with the proof I have from this biography, census records, family naming rituals, etc. I think I can safely conclude that the missing female Mitcheltree is my 5th great-grandmother. She had at least two sons, William R. Elliott, Esq. and his brother James. But whether she died here in Pennsylvania or back in Ireland is unknown, as my family has no records of William's parents in America.