Hi Maria,
Thank you for your message. I hope that someone with information contacts you and helps you to further your research.
I found a record of PAtrick Delargy in Griffith?s Valuation, in 1861 Layd:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=55&Submit.y=12&Submit=Submit&familyname=Delargy&firstname=patrick&baronyname=&countyname=ANTRIM&unionname=&parishname=
Do you know what religion the family were?
The civil parish of Layd falls under t two Catholic parishes; Layde and Cushendun. Records for Layde start in 1838, so if his baptism was registered there, there should be a record:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Antrim&parish=Layde&churchid=383
They start as early as 1834 in Cushendun:
http://www.irishtimes.com/ancestor/fuses/rcparishmaps/index.cfm?fuseaction=showidrecords&CityCounty=Antrim&parish=Cushendun&churchid=409
You can see from the above links where there are copies available. Alternativley you could try writing to the local parish priests for some assistance.
Most surviving Church of Ireland records are still held by the local clergy, although some are in the National Archives of Ireland and others are in the Representative Church Body Library in Dublin. Here are their websites: http://www.nationalarchives.ie/ and http://ireland.anglican.org/about/42
Lists of these surviving registers can also be found at the National Library of Ireland.
Somer other websites that you may find useful are:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Irish Times: http://www.irishtimes.com/ancestor/index.htm
Irish Genealogy Tool kit: http://www.irish-genealogy-toolkit.com/
Genealogy Links: http://www.genealogylinks.net/uk/ireland/antrim/index.html
From Ireland: www.from-ireland.net
North Antrim: www.northantrim.com/northantrimgenealogy.htm
Historic Graves: www.historicgraves.ie
Best of luck,
Genealogy Support