The Azzurri have finished with the wooden spoon on nine occasions but defeated Scotland in the final round last season to end the competition fifth.
Under new coach Jacques Brunel, who led the team for the first time last year, Italy have attempted to play a more creative brand of rugby.
In the autumn that approach saw them defeat Tonga, push New Zealand close for an hour before succumbing 42-10 and come within a missed penalty of beating Australia.
And after displaying their ability to compete at times with the best, No.8 Parisse has challenged his Italy team-mates to display more than just moments of magic.
"We showed that if we played our best then we can beat teams like France, or Wales or Ireland," he said.
"We were really close to beating Australia in the last Test in November so it is good for us to see that we have improved and we are playing better.
"We are really close to a lot of nations but it is also great to get results so in the Six Nations we have the chance to play three times at home against France, Wales and Ireland and we want to try and take this opportunity to play good rugby but also get results.
"In the past, even three or four years ago, we did great matches but the match after we would come back and do a lot of mistakes.
"That is the strength of the big teams in the world, to every time and every single match play your best and be competitive."