If it's any help, the parish is Ballybrack (my grandmother was born to an unmarried mother in Loughlinstown in 1923). It's the Church of St. Alphonsus and Columba, Church Avenue, Killiney -
http://www.ballybrack-killiney-parish.org/.
I ended up taking the Ancestry DNA test to try and find grandmother's birth parents - it's been an interesting journey

Out of curiosity - did you look up the birth record on irishgenealogy.ie? I had some luck for the father of an illegitimate great uncle where the father was listed on the cancelled cert on the same page.