Parrott's brace was an immediate response to Marc Vales' 51st minute opener as Jason Knight and Daryl Horgan rounded off the victory, Stephen Kenny's first victory as Ireland manager.
Ireland were poor in the first 45 minutes and only fashioned one chance of note as a wonderful passing move resulted in Josh Cullen's cross to James Collins who flashed his header wide.
The second-half got off to the worst possible start as Vales rose highest to head home a right-sided set-piece. It was the moment Ireland needed to spring into life.
The 19-year old Parrott found space and drove at the Andorran defence, turned to create space and found the bottom corner to score his first senior international goal in the 58th minute.
He had his second international goal just moments later. Conor Hourihane, advanced on the right side, dug out a cross to the far post and the keeper couldn't keep out Parrott's powerful downward header in the 61st minute.
Ireland had found their groove at this point and would go on to score a further two goals in the final 10 minutes.
Substitute Daryl Horgan found an inch-perfect cross to Knight who deceived the defender at the back-post to tap home his first international goal six minutes from time.
Knight would go on to return the favour four minutes later as his fantastic cross, this time from the right, found Horgan and his diving header found the net to round off an impressive victory.
"We were disappointed with the first-half and to go behind, but the response from that point was really impressive," said Ireland manager Stephen Kenny.
"I'm delighted for Troy. He has had a tough season but we believe in his talent and ability, and he took his two goals brilliantly which told us a lot about his character.
"Hungary will be a much tougher test so we'll have to raise our performance levels for that match on Tuesday but we're delighted to come away with a win."
Andorra: 12 Iker (GK) (13 Francisco Pires (GK) 76), 3 Marc Vales, 4 Marc Rebes (24 Christian Garcia 72), 8 Marcio Vieira (C), 10 Ludovic Clemente (2 Cristian Martinez 59), 14 Jordi Alaez, 15 Moises San Nicolas, 16 Alex Martinez (6 Ildefons Lima 76), 17 Joan Cervos, 19 Ricard Fernandez (9 Aaron Sanchez 59), 20 Max Llovera.
Subs not used: 1 Josep Gomes (GK), 5 Albert Alavedera, 7 Marc Pujol, 11 Sergi Moreno, 21 Marc Garcia, 22 Victor Bernat, 23 Eric De Pablos.
Republic of Ireland: 1 Gavin Bazunu, 2 Matt Doherty, 20 Dara O'Shea (4 Shane Duffy 86), 5 John Egan, 11 James McClean (3 Ryan Manning 86), 6 Josh Cullen, 8 Conor Hourihane (12 Harry Arter 86), 10 Troy Parrott (18 Jamie McGrath 82), 7 Ronan Curtis (15 Daryl Horgan 66), 14 Jason Knight, 19 James Collins (9 Adam Idah 66).
Subs not used: 16 Caoimhin Kelleher (GK), 23 Mark Travers (GK), 17 Jayson Molumby, 21 Daniel Mandroiu, 22 Chiedozie Ogbene, 24 Lee O'Connor.
Referee: Xavier Estrada Fernandez (ESP)