Parrott's 31st-minute tap-in from the excellent Aaron Connolly's inch-perfect cross was enough to seal a second win of the Group One campaign in as many games in front of a crowd of 3,658 at Tallaght Stadium.
Tougher tests lie ahead for Stephen Kenny's men, but a maximum six points from their opening fixtures against Luxembourg and the Armenians is just what was required.
Ireland might have taken the lead within 16 minutes when Norwich striker Adam Idah found himself in a promising position in front of goal, although he dwelt on the ball and keeper Sevak Aslanyan pounced to deny him.
However, there was nothing Aslanyan could do to deny Parrott 14 minutes before the break when Brighton's Connolly worked his way into space down the left before delivering the perfect cross for the Spurs man to tap home.
Connolly himself might have doubled his side's advantage as the home side finished the half strongly, but they went in at the break just a goal to the good.
The home side created further chances after the break, but were unable to convert any of them to leave the visitors with hope, but they eventually saw out the time without mishap to claim the win.