The Irish Consulate in New York explained that the parent would need to be registered in the "Foreign Birth Register" which is held at the Consulate, effectively a listing of those of Irish citizens born abroad who are entitled to Irish citizenship who have their births "registered."
Once you have established your Irish citizenship – which can take up to 18 months to process – you can apply for an Irish passport. This can take up to six weeks to process and you can do this through your nearest Irish consulate.
And then, you too can sashay through the EU passport section at Dublin airport after the overnight flight from New York!
*Originally published December 2013