Amauri followed up his goal against Torino on Monday with a brace as Roberto Donadoni's side stretched their unbeaten run to nine games and moved up to seventh in the table.
Second-bottom Livorno, on the other hand, have lost seven of their last eight games and not won for over two months.
The visitors took the lead after only two minutes, when former Livorno loanee Raffaele Palladino arrived at the far post to volley home a Jonathan Biabiany cross.
Amauri doubled Parma's lead four minutes from time and the striker was on target again from the penalty spot after Emerson fouled Pedro Mendes on a miserable afternoon for Livorno.
In the other Serie A game to be played on Saturday, Davide Ballardini's new side Bologna held his former club Lazio to a goalless draw.
Ballardini this week replaced Stefano Pioli on a short-term deal until the end of the season and got up and running with a point to ensure Lazio are still yet to win away from home in Serie A this season.
Bologna are now two points above third-bottom Sassuolo, who play Milan on Sunday.