The information that you seek can be found at familysearch.org. It is a free site, you just need to join to search.
Here is what I found for Peter Fauci. https://www.familysearch.org/search/record/results?q.givenName=Peter&q.surname=fauci&q.birthLikeDate.from=1905&q.birthLi…
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Dear Ms. Fauci-Wills,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the Population Schedules for the 1910 Census (not digitized for NYC); the Population Schedules for the 1920 Census (digitized for NYC); the Population Schedules for the 1930 Census (not digitized for NYC); and the Population Schedules for the 1940 Census (digitized for NYC) in the Records of the Bureau of the Census (Record Group 29) that may list Peter Fauci and his family and/or parents. For access to the non-digitized census schedules, please contact the National Archives at Washington, DC - Textual Reference (RDT1) via email at email@example.com.
Due to the COVID-19 pandemic and pursuant to guidance received from the Office of Management and Budget (OMB), NARA has adjusted its normal operations to balance the need of completing its mission-critical work while also adhering to the recommended social distancing for the safety of NARA staff. As a result of this re-prioritization of activities, you may experience a delay in receiving an initial acknowledgement as well as a substantive response to your reference request from RDT1. We apologize for this inconvenience and appreciate your understanding and patience.
The Census schedules have been digitized by Ancestry. In response to the COVID-19 pandemic, the National Archives has partnered with Ancestry to make the vast majority of their NARA-digitized holdings freely available to the public. Anyone with Internet access may create an account, access NARA records, and use other Ancestry resources, such as their educational offerings and family tree-maker application. For more information see Ancestry’s announcement -- “Free At-Home Education Resources From Ancestry® and Access to Nearly 500M National Archives Records”.
For birth records prior to 1910, marriage licenses for 1908 to 1949, and death certificates prior to 1949, please contact the Municipal Archives of New York City. The Italian Genealogical Group has indexed (via databases) many of these records.
For additional resources, we suggest you check the FamilySearch Research wiki for New York City Genealogy.
We hope this is helpful. Best of luck with your family research!