Hernandez did give Palmero the lead after six minutes when he headed a Andrea Mantovani pass into the back of the net.
Hernandez might have added a second after 13 minutes, but he was denied by a smart save from Mirante.
The home side continued to create the better openings as the half developed, and Franco Vasquez came close with an effort after 29 minutes.
Parma's best chance of the opening period well to Sebastian Giovinco after 38 minutes, but his strike from outside the box just drifted wide of the target.
Stefano Okaka Chuka passed up a glorious chance for the away side moments into the second period, but he struck wide.
It was Palmero who so nearly extended their lead after 48 minutes, however, when Matias Silvestre struck the bar with a powerful effort.
Parma levelled the match after 55 minutes, when Okaka Chuka headed home from close range.
The away side continued to carve out the better openings, and took the lead after 70 minutes when Jonathan Biabiany struck home from inside the box.
Hernandez remained the home side's biggest threat in the latter stages, and came close with an effort after 81 minutes, but it was Parma who held on for all three points.