Was your parent already on the Foreign Births Register on the day you were born?
Where the Irish-born ancestor is a great-grandparent or further back, this is the whole question. Registration takes effect from the date it happens, not from birth, so your parent had to be registered already on the day you were born for anything to pass to you. If your parent's entry was accepted, the generations above them have already been checked. The date is printed on their Foreign Birth Registration certificate, and the Department of Foreign Affairs can confirm it if the certificate is lost.