Kath, I now see your problem, the Marsh Road is in St Marys Parish Drogheda County Louth but in the diocese of Meath, marriage records for here do not commence until 1871 for church records, civil records are from 1864 unless Protestant. Baptisms are much earlier.
To locate a record of your Mary can you give an approximate age, unfortunately we are dealing with common names.
I have found a few records of interest but cannot stand them up until we find a record for one of the siblings giving the mother. Prior to 1864 we are reliant on church records only by and large where they exist.
The only Rose Brady I found with father Patrick or Pat was born circa 1857 in Kells, Co. Meath about 30 miles from Drogheda, mother was a Mary Hinney or now Hiney normally. Found no other children to these parents on Roots Ireland but maybe I am using the wrong spellings and the variant search is not picking up. Oddly there is a Julia Brady born to a Michael Brady and Alice Hiney in Celbridge workhouse in Kildare, not a good thing. I wonder if two brothers married two sisters. There appear to be two related families in Drogheda. This area was very industrialised at that period especially around the Marsh area, docks, factories, linen etc.
I found a number of records that I think may be relevant but the dots do not join immediately for me. Firstly in 1882 a Julia Brady this time father Patrick of the Marsh marry in St Mary’s, she married a John Burns, interestingly the witness’s are Elizabeth Brady and a James Donegan, this surname occurs later again, Rose’s age is a movable feast in the census, aging 19 years in ten, not unusual as the pension came in circa 1906 and being old was helpful, no records prior to 1864 whereas in England civil records commenced in 1837. Why I say this is a Rose Brady is dead in 1950 in Dundalk age given as 80 present at death is a Julia Donegan. (I cannot find suitable marriages or births in Ireland but Drogheda had a regular boat service to Liverpool, more than weekly).
Also found a death of a Mary Brady in the Marsh attended by a Bridget Brady in 1890 I think but she was already a widow so it may be Bridget was the wife of Patrick, have you used the site Irish Genealogy which is free but sign in, civil records mostly there but not all certs visible as yet, births to 1915 I think and deaths to 1965 but full records not all the way back. Same for marriages.
https://www.irishgenealogy.ie/en/
When you supply Mary’s age I will endeavour to locate her birth, the graves in St Peters are online and I will see if I can locate the site, I have the records ofr Calvary and Cord will check later.
I am attaching some records for your perusal, note a Mary Brady spinster dies in Ardagh, outside Drogheda but a brother Patrick Brady was present but again could be another Brady family.
Regards
Pat