Parish Court and First Judicial District Court
The Parish Court handled a wide variety of cases including those relating to family issues, among them suits for separation or divorce. Records of this type, of course, are very useful to genealogists, since the suits will include the names of both spouses, reference to the date and place of marriage, the names of other relatives, and possibly even the names of future spouses (depending on the grounds for separation/divorce).
The First Judicial District Court (sometimes referred to simply as the District Court or as the First District Court) had original jurisdiction in Orleans Parish and appellate jurisdiction in St. Bernard, Plaquemines, St. Charles, and St. John Parishes. In 1825, newly-created Jefferson Parish was added to this list (prior to that year, Jefferson Parish was part of Orleans--an important fact to keep in mind when researching families and properties just upriver from the Crescent City). Divorces and separations can also be found in the First Judicial District Court, along with other types of civil suits, including a good many slavery-related cases.
Indexes to the earlier records (1813-ca. 1835) for both of these courts are available in NUTRIAS' Archival Inventories page.