William Byrne and Betty Dunne, Walsh Island

I found the baptism record of my 2nd great grandfather, William Byrne born March 1824 in Walsh Island, Offaly. His parents were William Byrne and Betty Dunne but I cant seem to find any info about them. I cant find their wedding, death or any baptisms for siblings of William Jr. anywhere.

William Byrne b:1824 married Anne Forde, (b:1836 Coolelan) in Coolelan, Rathangan, Kildare in Feb 1855. They had 2 children in Coolelan; Jane Byrne b:1858 and Hugh Byrne b:1860 before emmigrating to New York City in January 1861.

Most likely his parents William Byrne and Betty Dunne remained in Offaly so it's them and their families that I'm concerned about. I've searched rootsireland and other sites for Byrne/Dunne in the Walsh Island/Ballykean area and found plenty but I just cant connect them yet.....I'm sure both William Byrne and Betty Dunne had siblings nearby.

I have William Byrne b:1824 New York City death certificate from 1896 that states his parents names as William Byrne and "Nettie" Byrne. Would a Betty or Elizabeth be called "Nettie"?...doesnt seem right.

I could use some help,

Thanks,

Tim Byrne