Welcome to History Hub found a naturalization for Anna on familysearch.org 1930 but nothing for Paul.
Their arrival at ellis island Anna is listed as Gio Battista Scarrone.
Paolo Barbiroglio, "New York Passenger Arrival Lists (Ellis Island), 1892-1924" (familysearch.org)
marriage record 1909
Is this them?
Paul & Anna Barberola in Boonton, Morris, New Jersey
You could view this on Ancestry.com, or here is the link on FamilySearch.org (you would need to register for a free account).
It is possible this is them but According to the ship documents Paul headed to Glastonbury to a farm upon arrival then when Anna came they got married in NY but were back on the farm by 1920. That with the last name being misspelled and her first name misspelled I'm just not sure its them.
Dear Ms. Kaminski,
Thank you for posting your request on History Hub!
We suggest that you search city directories for the years 1909 and 1910 which may provide more details about the residence location of your grandparents. The New York Public Library Digital Collections has searchable city directories for the New York City area and the Library of Congress has directories for Connecticut. You may then use that information to refine your search of the 1910 US Census.
You may wish to search Ancestry.com, FamilySearch.org or Fold3.org for the city directories. There may be a fee for using Ancestry or Fold3. Instead, please check for access at your local library as many library systems subscribe to these sites, making them free for their patrons.
Since you have already submitted an index search request to the US Citizenship and Immigration Services (USCIS) Genealogy Program for records of your grandparents, we suggest that you wait for those results. If they did not complete the United States naturalization process, USCIS should be able to provide the certification on non-existence (see FAQ section), and although they won’t have naturalization records, it is likely that INS did create a file that the Italian consulate would require as part of the dual citizenship process.
Please note that USCIS Genealogy does NOT provide certification of files in their holdings, therefore you will need to keep all correspondence (including envelopes) in relation to their request.
We hope this is helpful. Best of luck with your family research!