I believe that the prefix "para" is Latin for "next to"; in he case of the term "paramedic" this means next to a doctor, or "medic". In the US military the term used to refer to medically trained paratroops who would parachute into the scene of injured military personnel, and later, into civilian emergency scenes like airplane crashes or injured persons in the backcountry.
Nowadays the military is not as involved in civilian emergency medical work (Coast Guard excepted) due to the large number of civilian medevac helicopter units we have, but at times the military still assists with search and rescue.