You should do whatever genealogy and obtain whatever documents you can that show you are clearly descended from an Italian person. So far these are all documents located in the United States of America.
Let's go over them once more.
Your birth certificate.
Your parents marriage certificate - if in the U.S.
Their death certificates - if in the U.S.
Their birth certificate - if in the U.S.
Naturalization Papers
Ship Records/ Immigration
Census Records - if at this point 1930 or before.
In your final quest to obtain Italian Citizenship you may not need to submit an entire genealogy, but the more you have the better at this point. You may encounter some "missing" pieces.
C 2020 Ancestry Worship - Genealogy BlogSpot
