You first need to determine the civil parish of birth. Once you know that, then you can focus on the archdiocese or diocese where the birth might have been recorded. The Archdiocese of New Orleans as well as the several diocese in other parts of the state all encompass specific civil parishes. In Louisiana, a civil parish is the same as a county, not to be confused with a church parish. This site should direct you the appropriate repository:
Click on "Geographic Location", then "United States" and then "Louisiana". It is all explained and contact info is also included. If you don't know her parish of birth then I guess you can try a shotgun approach and query all of them. However, I would try to pin down her place of birth first (if you don't already know it). Constance gave you a lead with the SSDI info. You can always order the SS5, it should give her place of birth plus parents' info (hopefully). A death certificate might also furnish the place of birth but I don't recall offhand what restrictions New York places on obtaining one. That can be easily checked.
It's necessary that you be real clear about the surnames. You give her father as Giuseppe TORTOROSA(?) and her name as TAURGRASSO (Margaret). So, was her maiden name the former and is it correct? After all, that is how it will be recorded in sacremental records. I hope this isn't too confusing. Rick