Different government bodies can pass valid laws that affect the same cemetery, as long as those laws do not conflict with one another. For example, the Louisiana Cemetery Board may license the owner of a cemetery and ensure that it follows state law. That same cemetery may be on the National Register of Historic Places (federal law). Vandalism in that cemetery is a state and municipal violation.
Native American Graves Protection and Repatriation Act
The National Historic Preservation Act
Every parish is special, and will have local laws that reflect this fact. Bossier Parish has just a few code sections on cemeteries, but in one of those the police jury declared a list of 76 rural cemeteries in the parish as historical sites. New Orleans has 64 separate code sections on cemeteries. Grosse Tete has but 2 sections on cemeteries.
Generally speaking, if there are no current problems, a parish will have little need to pass laws concerning cemeteries. If there are problems such as vandalism, abandonment, or theft, then a city or parish will take action. You can check your local municipal code here: municode.com.
Louisiana has cemetery laws spread throughout the Revised Statutes: Here are the main places to look:
R.S. 8:1 et seq. The Cemetery Chapter
R.S. 25:931 et seq. The Louisiana Historic Cemetery Preservation Act
Francis Norton 2018
In addition to the laws of government bodies, a cemetery authority can make up rules for the cemetery, including rules on hours, upkeep, and what type of markers are allowed.