The name of the parish and availability of records depends on the dates involved... parishes and churches changed over the years.
St. Mary's (name of the church & parish) is the current parish church in Donnybrook. Note there are other parishes and churches in Dublin city & county named St. Mary's.
Historic records for most Catholic parishes are available in the National Library. These cover up to about 1880. For more recent records you have to write to or phone the church to see if they are in a position to search for you (sometimes there's a fee involved).
For more recent records it might be simpler to search civil records. A BMD index is available on FamilySearch which covers up to 1958 for the 26 counties. Birth records after the late 1920s include mother's maiden surname which can help find possible matches for children as long as the surnames are not too common.
Ballsbridge Ave. is still there - it's located to the north of Merrion Road near the River Dodder, at Ball's Bridge.