Dear Jan Morris,
Thank you for posting your question on History Hub.
The Federal Government does not ordinarily create or maintain birth, adoption, death, marriage or divorce records. Such records are made and kept by State and local governments rather than the National Archives. We suggest searching substitute record sources such as probate records, church records, and newspaper records that may uncover more information about the parents of Mary Ann Greer.
The vast majority of records in our collection are not name-searchable. For more information on conducting genealogical research within our records, please visit our webpage: https://www.archives.gov/research/genealogy.
We suggest that you review the FamilySearch Research wiki for Arkansas Vital Records (https://www.familysearch.org/wiki/en/Arkansas_Vital_Records ). We also recommend contacting the Arkansas State Archives (https://www.arkansasheritage.com/arkansasstatearchives/home ) to inquire if they have additional family records.
Finally, a great resource for learning about, and searching for the records of those who served in the Civil War, check out https://www.archives.gov/research/military/civil-war.
Being that your ancestor was in Arkansas, I would focus your research on the records of Confederate Soldiers by looking into pension records (https://www.archives.gov/research/military/civil-war/confederate/pension ).
We hope this information is helpful in your research.
The National Archives and Records Administration