Niyi Adeolokun got two touchdowns and Kieran Marmion also scored for the Irish side, who are the competition's top try scorers with 42 this season.
Johan Meyer, Kelly Haimona, Dries van Schalkwyk and Federico Ruzza responded with tries for the hosts.
Connacht had never lost to Zebre and this was their 12th win over their Italian opponents.
But Zebre pushed them all the way in Parma and were rewarded with a try-scoring bonus point.
Craig Ronaldson and Haimona exchanged early penalties before Connacht wing Healy claimed the first of his three tries after good work by Bundee Aki and Tiernan O'Halloran.