The Verdict: Clear Path
With a parent born on the island of Ireland, you have been an Irish citizen since birth. Apply for a first Irish passport with your birth certificate and your parent's.
Citizenship passes automatically from a parent born on the island to a child born anywhere, as long as the parent was an Irish citizen, or entitled to be one, when you were born. There is nothing to register and no deadline to miss.
You need your civil birth certificate naming your parent, your parent's civil birth certificate showing their birth on the island, and any marriage or change-of-name document linking their names. Originals, not photocopies.
Two checks. If your parent formally renounced Irish citizenship by lodging a declaration of alienage before you were born, nothing passed to you; taking another nationality is not renunciation, so this is rare. And if your parent was born on the island on or after 1 January 2005, their own citizenship depended on their parents' status, so confirm it first.
If the parent you are counting is an adoptive parent, citizenship comes through the adoption rather than from your birth: go back and choose the adoption answer instead.
Your children born abroad can claim through the Foreign Births Register at any time, because you are already a citizen.