Hello Nancy, I see your baptism on the 28th October 1852 in St Peters Parish, Drogheda, this is in county Louth and the diocese of Armagh, Drogheda is in Louth north of the River Boyne and partly in Meath south of the river, St Mary's Parish is in the diocese of Meath but in both counties if you follow. The mothers name is Anne Heaney (Heeny)
I see no marriage record for the couple which I take to be indicative of the marriage occurring in St Marys where church marriage records are only available from 1870, there are civil records which commence in 1864 but obviously too late.
You can see the one line entry on the church record here https://registers.nli.ie/ The God Parents are a Bridget Heany and Nich. Andrews.
I see a Mary Teirnan (spelling as transcribed) baptised in 1854 on 25th September, Godparents Jane Clark and Patrick Branagan, parents as above, same again for a Patrick Joseph Teirnan, baptised 20th May 1856, God Parents Joseph Lonigan and Mary Anne Heaney. Also Catherine Teirnan baptised 7th Sept 1858, Godparents Catherine Conway and Thomas Green.
Another one
Name: Agnes Tiernan Date of Baptism: 13-Mar-1863
Parish: DROGHEDA Co. Louth
Father: Patk Tiernan Mother: Anne Heeney
Also
Name: Thomas Augustine Ternan Date of Baptism: 18-Feb-1865
Address: Parish: Drogheda Co. Louth
Father: Patrick Ternan Mother: Ann Heeny
Informant 1: Mary Kate Curran
The civil record is there for this birth but I am unable to see it as the linked page is to the wrong one, however it appears to be that this child died, have not checked the death register but there is another Thomas born to them in 1866 and the father is a farmer in the Twenties, a place on the north side of Drogheda, you can find the ledger entry here, you should also be able to find the parents deaths and any other children, it is free but sign in etc and click on civil, the church records are not here for Louth. https://www.irishgenealogy.ie/en/ click on civil and away you go.
You can see the location of the Twenties here https://www.townlands.ie/louth/ferrard/drogheda-st-peters/st-peters/twe… if the link does not work you can see the site here https://www.townlands.ie/
A land survey shows a Patrick Tiernan in the Twenties in 1854 and a neighbour Patrick Heany, if you work this site properly but with great difficulty you can find the actual plot the were tenants on at the time, I see a number of Tiernans having land in the area including a Nicholas and an Agnes which are names in the family above. You can drag the cursor from the 1854 to the modern map but difficult to see exactly where you are, these maps can be viewed on http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameS… also places can be checked, the main map office is here and you can view also https://www.osi.ie/
There is a site here about Louth http://www.jbhall.clahs.ie/ there are a number of Drogheda Facebook pages apply to join, Drogheda Sown Memory Lane, this is very active, Drogheda Down Memory Lane - Lost Family, not very busy and Drogheda Old Photos and Family Roots Finder but not a busy page. There is also a Louth genealogy page.
The area is now getting developed so may have changed a lot since then as the town expands.
Good Luck
Pat