The result lifts Rolando Maran's men out of the bottom three and above Cagliari, while Parma are now 12 points from safety at the foot of the table.
Roberto Donadoni's men were playing their first match in front of Giampietro Manenti, the club's third owner of the season.
The tension was obvious and the hosts picked up five first-half bookings, two of them in the space of five minutes for Daniele Galloppa, who was sent off after half an hour.
Chievo were comfortable with a man advantage and scored the decisive goal in the 54th minute, Zukanovic curling a wonderful 20-yard free-kick over the wall and into the corner.
The visitors should have doubled their lead seven minutes from time but substitute Ruben Botta scuffed his shot wide with the goal at his mercy after Riccardo Meggiorini's low shot had come back off the post.
Earlier in the day, club president Manenti had spoken of a contingency plan should they be relegated to the Italian second tier.
With the club now looking all but certain to drop down, Manenti also reflected on the mood among the players.
"I found a team of great professionals who will fight to survive until the math tells us otherwise," he told reporters.
"But if we drop to Serie B, we have a plan ready and in no case would that change the project.
"We've met the lads and they didn't give the impression of being resigned to their fate.
"I say to the fans: Trust me. Because what we are doing we are doing with a plan."
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