On a night when 17-year-old Tottenham striker Parrott was handed a senior debut for his country by Mick McCarthy – just as Robbie Keane had been back in 1998 – it was three more senior colleagues who opened their scoring accounts in the green shirt to clinch victory.
Blackburn defender Williams headed Ireland level after Callum McCowatt, representing the Republic at senior level for the first time, had given the visitors the lead, and Preston striker Maguire and Sheffield united counterpart Robinson struck after the break as they eased across the finishing line.
McCarthy's main focus was understandably on Monday's decisive Euro 2020 qualifier against Denmark – only Alan Browne survived from the side which lost 2-0 in Switzerland last month – but there were pluses for him other than the result against an All Whites outfit which was playing for the first time in 525 days.
Shamrock Rovers midfielder Jack Byrne turned in an encouraging display in central midfield, while skipper Robbie Brady warmed to his task on his return to the international fold.
If most eyes were on Parrott as the game got under way, it was Byrne, making his first start for his country, who grasped it by the scruff of its neck early on.
Seemingly ever present in the middle of the field, he found space and picked passes to stretch the All Whites' defence.
New Zealand skipper Winston Reid, playing his first game since March last year after a serious knee injury, left Parrott in little doubt as to what he thought about his challenge from behind, and fellow defender Michael Boxall was similarly unimpressed after the teenager went to ground inside the penalty area under his 18th-minute challenge.
However, New Zealand took the lead with half-an-hour gone when midfielder Ryan Thomas picked out Liberato Cacace in space on the left after playing a neat one-two with Chris Wood, and the defender sent in a low cross which McCowatt turned past goalkeeper Kieran O'Hara as he slid in.
The Republic got their reward when Williams met Brady's corner with a powerful downward header to level with seconds of the first half remaining.
O'Hara had to react smartly to beat away Thomas' deflected shot four minutes after the break, but Ireland took the lead three minutes later when, after Williams and Parrott had caught McCowatt in possession deep inside his own half, Maguire dispatched a sweet strike past Stefan Marinovic to finally open his account.
Marinovic raced from his line to deny Parrott after Byrne and Maguire had capitalised on a poor defensive header by Joe Bell, but 19-year-old debutant Lee O'Connor had to head clear at the other end after O'Hara could only parry Thomas' fiercely-struck 61st-minute effort.
Parrott's big night ended with 63 minutes gone when he was replaced to warm applause by Robinson, and the newcomer marked his 11th cap with his first goal when he headed home from O'Connor's inviting cross.
Goalkeeper Mark Travers made a fine late save to prevent fellow substitute Michael McGlinchey from reducing the deficit on a comfortable night for McCarthy's men.