Antonio Conte's men are considered an outside bet to triumph in France and face a stern opening test against Belgium in Group E.
Parolo concedes Italy are short on headline players who are considered world class, but is confident the right attitude will win the day.
"With the right spirit, anything is possible," the Lazio player said. "We do not have any top talents at their peak, but we do have players who can resolve a game.
"Many players have established themselves in both Serie A and the Champions League through hard work and sacrifice. If we do badly, you will write that we are second rate, but sometimes it is more beautiful to be a surprise.
"Going into the 2014 World Cup in Brazil there were some big changes in the squad. In this case we worked as a unit, seeing as those in the national side have been part of it for a while.
"We do have some good talents in the Under-21 side, although at times there are too many expectations piled upon them. Abroad, players are given more time to grow. There might be a small generational gap, but we'll bridge it soon enough."
Coach Conte will head to the Premier League to take charge of Chelsea next season and Parolo sung the 46-year-old's praises.
"The coach always has energy and strength," he added. "He has a winning mentality that spurs you on to give everything. You see this in training. There is a great group spirit and desire to make sacrifices for everyone."
After taking on Belgium, Italy – who meet Finland in a friendly on Monday – face Sweden and Republic of Ireland.