I found quite a few records on FamilySearch for Patrick which I added to his profile. That will make it easier for everyone to help. They can see what is known.
The maiden name of his wife is "Kane", according to child Lawrence's death registration;
https://www.familysearch.org/ark:/61903/1:1:2WTX-WZ4
In the 1865 census their niece, Mary Ann Kein (Kane) is visiting. Census is attached to Patrick.
The best source to find Patrick's parents would be his marriage record or death record. It would also be helpful to identify when he immigrated.
This is interesting, a birth record for a Mary Casey b. 1868 in Ireland. Looks like the same family. This child is not on the 1870 NY census though. There is a Minnie Casey under 12 months. Were they back on a visit and the child perhaps did not survive.
http://"Ireland Births and Baptisms, 1620-1881", database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FRQ7-TMP : 15 February 2020), Patrick Casey in entry for Mary Casey, 1868.
And another child Patrick b. 1870.
https://www.familysearch.org/ark:/61903/1:1:FRSB-KGQ
Cant be the same family??
I added the Ireland Family Brick Wall category to Patrick's profile, so hopefully that gets more attention from the Ireland volunteers.
I also added a link to the FamilySearch tree that you have. This will hopefully save people giving you info you already know.