According to what I read in the census records Thomas came over 1866-1867 and Henora came over in 1870. They were also married ca 1872.
They were in Pittsburgh in the 1880, 1900 and the 1910 census. The 1880 census is the most telling of all. In addition to their daughter Margaret 8, Mary 7, Ella 3, Annie born Jan 1880, Thomas' father James Moriarty 65 is also listed. I could not find James or Thomas in the 1870 census.
Living next to Thomas and family in 1880 was the family of Thomas 35 & Johanna 30 Haley. Most likely more than just coincidence. This same family is in the 1870 census in Pittsburgh under the spelling of Healy.
On April 12, 1867 Jas Mariarty 60, Mary 55, Thos 29 arrived in New York on the "Manhattan". Also listed was a Johanna Moriarty 23 but she was not listed with the rest.
On Aug 23, 1870 Honora Haley 15 arrived in New York on the "City of Washington".
Some of these records you can look at for free at labs.familysearch.org and Ancestry.com. For those you cannot see for free I suggest checking with your local libraries to see if they have Ancestry or Heritage Quest.
If you haven't already I suggest getting Thomas and Henora's death certificates from the PA Div. of Vital Records. They should indicate the parents' names. Also try to get their obituaries. At the very least they should indicate siblings' names, what church they were members of and/or the cemetery they are buried in.
The cemetery may have info on them and quite possibly some relatives who were buried in the same cemetery. I believe there is an archives for the Pittsburgh Archdiocese. They may be able to help you with baptism, marriage and death records.
You might also try the Western Pennsylvania Genealogical Society in Pittsburgh. They might be able to help with the obits.
When you get stuck track down the info on Thomas Haley. Assuming he was Henora's brother his obit or death certificate may have the info you need on their origins and parentage.