The Azzurri had to battle hard against the hard-working underdogs, which will have encouraged the Irish, but they came away with a 27-10 win to leave them level on points with Australia in Pool C.
Assuming the Wallabies beat minnows Russia on Saturday, that will leave one more qualifying place for the winners of Sunday's clash between the two RBS 6 Nations sides in Dunedin.
Parisse, who scored the opening try in the third minute, told ITV4: "We are happy because we took the five points - that is what was important for us.
"It was a lot of work but we scored four tries - that is the important thing. So we can look ahead to the Ireland game with confidence.
"It will be a big match. We have just five days to recover but we are looking forward to it with a lot of confidence."
Coach Nick Mallett is aware of the challenge presented by Declan Kidney's side, particularly after their stunning 15-6 win over Australia, but is hopeful his impressive front row of Martin Castrogiovanni, Leonardo Ghiraldini and Salvatore Perugini can again play their part as they did against the States.
Leicester prop Castrogiovanni was another try-scorer today along with fly-half Luciano Orquera, with a second-half penalty try confirming the crucial bonus point.
Mallett said: "Ireland are a side who really played so well against Australia, so we've got to play really well.
"I'm hoping Castro, Leo and 'Doctor' Perugini can put us on the front foot.
"At the scrum today, we dominated completely. The game was not as open as the spectators would have liked but you have to play to your strengths and we deserved the penalty try - if not a couple more, quite frankly."
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