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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.