Italy's 23-18 success at Stadio Olimpico blew the competition wide open, with France and reigning Six Nations champions Wales both beaten on the opening weekend.
"We are really happy with the victory," try-scorer Parisse said on BBC1.
"We talked a lot about our ambition as a team before the tournament, but when you talk and you don't get the result, you don't have any credibility.
"Today, we have won against France, a team that played very well in November. For us, it is a fantastic beginning to the Six Nations.
"We kept the ball for a lot of phases, and I am really happy because we played together so much. I think we really deserved the win."
Leicester prop Martin Castrogiovanni scored the decisive try of a thrilling encounter, but Italy fly-half Luciano Orquera ran the show majestically.
Orquera kicked a drop-goal, two conversions and a penalty before his replacement Kris Burton's late drop-goal saw the Azzurri beat France for a second successive Six Nations occasion on home soil.
France conjured first-half tries for number eight Louis Picamoles and wing Benjamin Fall, with Frederic Michalak slotting two penalties and a conversion, but they could have few complaints about the result.
Italy survived a frantic finale, holding out despite being under sustained scrum pressure after replacement hooker Davide Giazzon was sin-binned two minutes from time.
But they had heroes everywhere as France threw everything at them, gaining a victory that means they will head to Edinburgh next weekend as favourites to beat Scotland.
Castrogiovanni said: "We couldn't have started the tournament better than this, repeating what we did (against France) two years ago.
"France are one of the best teams and one of the favourites to win the tournament, so for us it is a huge start.
"We still have other four games to go, and I hope we keep playing like that. If we play like that, I think we can have a chance against any team.
"Our defence was unbelievable, and against any team in the world, if you put them under pressure with your defence they can't carry the ball forward.
"It's not how you want to finish a game, with seven men in the scrum five metres from your own line, but the boys really wanted it."
France must now pick themselves up for next Saturday's appointment with Wales in Paris, having seen their title hopes dealt a grievous blow.
"We need to congratulate Italy. They played very well, with a lot of passion, and kept the ball well," France coach Philippe Saint-Andre said.
"We had control of the game when we were five points in front, but we lost the ball too many times and were not precise. When you are not precise in the Six Nations, you are punished.
"The players are very, very disappointed. We prepared well for the game and we respect the Italian team. They improve each year, and in November they were unlucky to lose to Australia.
"Wales against Ireland was a fantastic game yesterday. Wales are very quick and physical, and we need to prepare very well and we need to win. It is not acceptable to lose at home."
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