Parker was a naval officer of considerable prewar experience who had also been an instructor at the U.S. Naval Academy. After some initial service in the James River/Hampton Roads area (during which he witnessed the Battles of Hampton Roads), he became the commandant of the Confederate Naval Academy, for which he personally wrote many of the texts used in the curriculum. His memoirs, Recollections of a Naval Officer, are considered a classic (and are a good read-- Parker was an interesting and frequently entertaining writer), and has been reissued a number of times by the Naval Institute Press.
His was a literal brother-vs.-brother situation, as his father and brother, both named Foxhall and both also naval officers, remained with the Union. According to William, Foxhall (the younger) was initially the secessionist and William the Unionist, and they argued and ended up each convincing the other. (!)