Hungary’s victory in Yerevan earlier on Thursday simplified the situation for the Republic of Ireland.
Knowing defeat would end their qualifying hopes, they needed to produce the best performance of the Hallgrimsson era against the 2016 European champions.
And while Evan Ferguson’s injury robbed the Irish of their top scorer, Parrott stepped up in spectacular fashion.
The AZ Alkmaar forward, who has six goals in seven Eredivisie games this season, led the line with energy and intensity, forcing Portugal keeper Diogo Costa into a mistake while attempting to clear the ball after 15 minutes.
From the resulting corner, Jack Taylor found Liam Scales at the far post, with the Celtic defender’s header back into the six-yard area turned in by Parrott.
Sensing opportunities against a shaky Portuguese backline, Chiedozie Ogbene also looked threatening for the hosts and nearly doubled the lead when he cut inside and struck a post with a curling right-foot shot.
With seconds remaining in the first half, Parrott handed the Irish their first two-goal lead in two years when he cut inside Ruben Neves and beat Costa at his near post, sparking ecstatic scenes among the home fans in Dublin.
The Republic of Ireland would have expected a strong response from the visitors, and while Caoimhin Kelleher produced a fine save to deny Goncalo Ramos, the hosts were able to comfortably see the game out and complete a famous win over the team ranked fifth in the world.
It is the Republic of Ireland’s first competitive win over Portugal in seven attempts stretching back to 1995 and a vital boost in their bid to reach the World Cup for the first time since 2002.