Donadoni committed to two more years of service with the Ducali club earlier this week, but it was only after Gianluca Sansone was dismissed for the hosts that Parma came alive.
Nicola Sansone broke the deadlock in the 72nd minute, before teeing up Amauri almost straight away, and the visitors sealed all three points in the final minute through Aleandro Rosi before Migjen Basha grabbed a consolation in injury-time.
The Crociati had drawn once and lost three times on their travels before today and it seemed their wretched run might continue as Torino started brightly.
Gianluca Sansone should have put them ahead but, despite being unmarked, he headed over from Alen Stevanovic's wonderful cross.
Head coach Donadoni had reportedly taken just three minutes to agree a new deal with Parma but he would not have been happy as Amauri missed an open goal from Jonathan Biabiany's cross from the byline.
Gianluca Sansone continued to cause problems for Parma's defence after the interval and saw Nicola Pavarini make a smart save.
However, the 25-year-old, on-loan from Serie B side Sassuolo, picked up his second booking for dissent in the 53rd minute.
While Parma could not immediately capitalise, they did take break Torino's resistance in the 72nd minute, with Nicola Sansone slotting home after being teed up by Marco Parolo.
The goalscorer turned provider two minutes later, with his cross finding Amauri to powerfully head into the net.
Rosi sealed the points for Parma in the last minute, tapping in from Biabiany's centre.
Basha salvaged some pride for the hosts, though, finishing off after some great individual work from Alessio Cerci but it was too little, too late.