I think I answered that on a previous post - to search for records it's the name of the parish you need to know. For Leeson Park. Avenue during the late 1800s and early 1900s RC parish would be Rathmines, Donnybrook or possibly St. Mary's Haddington Rd. - it''s located right on the boundary of these parishes. I think Rathmines is the most likely, but the other two are also possible, and RC parish boundaries are not as static as those for civil parishes and may have moved slightly over time.
Records were kept by the parish churches, and for these three parishes the main churches were, located at the Mary Immaculate, Refuge of Sinners Rathmines village, Sacred Heart Donnybrook village and at St. Mary's Haddington Rd. just south of the canal. All three are still parish churches.
Records for Rathmines and Haddington Rd. parishes are included on the IrishGenealogy website, and cover baptisms and marriages up to the very early 1900s. Records for Donnybrook parish are not online at the moment (**correction - see later post**).
RC parish registers for this timeframe for the city area do not usually include details of death/burials.