Searching for Parish baptism record

Ancestry gives the following death record for a miner named Cornelius Sullivan living where your Cornelius lived in 1870. This Cornelius is a bit older than the age given in 1870.
Name: Cornelius Sullivan
Birth Date: abt 1827
Birth Place: Ireland
Death Date: 13 Jul 1879
Death Place: Central Mine, Keweenaw Co., Michigan
Death Age: 52
Occupation: Miner
Race: White
Marital Status: Married
Gender: Male
FHL Film Number: 1310167

In the 1880 census, Mary Sullivan is a widow, with five sons. She has moved about ten miles south of Central Mine.

Name: Mary Sullivan
Age: 40
Birth Date: Abt 1840
Birthplace: Ireland
Home in 1880: Calumet, Houghton, Michigan, USA
Street: New Town Elm Street
Dwelling Number: 233
Race: White
Gender: Female
Relation to Head of House: Self (Head)
Marital Status: Widowed
Father's Birthplace: Ireland
Mother's Birthplace: Ireland
Occupation: Wash Woman
Months Not Employed: 12
Neighbors: View others on page
Household Members Age Relationship
Mary Sullivan 40 Self (Head)
Jamess Sullivan
14 Son
William Sullivan
12 Son
Cornelius Sullivan
10 Son
John Sullivan
8 Son
Timothy Sullivan
6 Son

According to Irish naming patterns, the first son is named after the paternal grandfather, the second after the maternal grandfather, the third after the father. Note that none of the children here is called Jeremiah. Therefore, I believe the baptismal record you located is probably not for your ancestor.

About Julia Moynahan--on Ancestry, I saw a marriage record for one of Cornelius's sons; he was married to Julia Moynahan.

Mary Sullivan's maiden name was Fitzpatrick, according to one of her son's marriage records on Ancestry.

I hope this is helpful..

Correction...the son was married to Julia McCarthy.  Sorry for confusion.

Julia, thank you soooo much for taking the time to look into this!! It is the info on Cornelius and Mary's children in Michigan that show/prove that this Cornelius is my G2. My dream is that someone has a family tree that will include 'my' Cornelius and his parents and ancesters. Thanks, again!

Don't give up, Tim.  Have you found Cornelius in census records before 1870?  Who was he living with? It was rare for an Irish person to emigrate by themselves.  Instead, someone they knew, usually a relative, had already emigrated. So when Cornelius first came to the U.S., he likely lived with family members.  You can trace such individuals and see where they hailed from; maybe you will find a connection to Cornelius that way.

Another idea. Have you seen birth, baptism, marriage, death records for all of Cornelius's children?  Sometimes the parents' places of origin are given in those records.  I recently found a baptismal record in New York that included the information that the parents were from Longford.

Have you seen Cornelius's obituary?  It might include his birthplace.

I hope this is helpful.

Thanks, Patricia. I will not give up. Thanks for the ideas.