Parish refers to the area for which a church is responsible. This dates back to medieval times, when the church parish would often cover the same area as the manor, as the church was often closely associated with the manor - many were built by the lords, and the priests were funded, as well as monks etc., to hold prayers for the lord or the manor, and to hold processions, as well as to organise the peasants.
In Louisiana the term parish is used as the equivalent of "county" or "township", a basic unit in the state's government structure.