Civil registration of births, deaths and Catholic marriages in Ireland began 1864 (from 1845 for other marriages). For earlier dates like you are looking for you need to search for church records (if they exist). To do so you need to know the family's religion and where they lived (parish if not actual townland- 'Ballymena' covers a large area and could refer to the town of Ballymena (in which there are many churches) or the wider surrounding area).
Are these two events something from LDS site by any chance?