Raffaele Palladino's early goal should have set the tone against a team who went into the game with only three wins and four draws from 18 games.
But instead Parma were made to withstand a second-half onslaught before Amauri's double in the last five minutes, including a stoppage-time penalty, wrapped up the points.
It seemed straightforward when, in the second minute, Jonathan Biabiany crossed from the right and Palladino finished with his left foot.
Livorno twice tried their luck from distance, Emerson shooting well over before Ibrahima M'Baye's effort flashed just wide.
Palladino went close to a second in the 17th minute, his shot brushing the outside of the post from Afriyie Acquah's cross.
Visiting keeper Antonio Mirante had to make saves from Leandro Greco's long-range shot and Paulinho's free-kick before half-time, while Amauri shot wide at the other end after being teed up by Biabiany.
After the break, though, it was almost all Livorno as the basement-dwellers sought an equaliser.
Luca Siligardi was denied by Mirante after a neat exchange with Paulinho and Marco Benassi shot over.
Piccini was denied at point-blank range by Mirante after meeting Siligardi's cross with a diving header.
Benassi, again, and Leandro Greco both shot wide, the latter from close range, before Mirante produced two more excellent saves to preserve his side's advantage.
Siligardi's 30-yard drive and Emerson's left-foot effort both extended the keeper but he proved equal to the task on each occasion.
And his efforts were rewarded as the home side's challenge petered out in the final 10 minutes.
Marco Parolo almost made it 2-0, only for keeper Francesco Bardi to match his opposite number's efforts.
Bardi also denied Nicola Sansone from close range but could only set up a goalmouth melee, in which Massimo Gobbi contrived to hit the post before Amauri finally converted.
And deep in stoppage time, Emerson fouled Mendez allowing the Brazil-born striker to score his second from the penalty spot.