Argentine Tevez struck in the 25th and 32nd minutes to go back to the top of the Serie A scoring charts with 18 goals and extended Juve's lead to 14 points after Roma had cut the gap to 11 with a last gasp 2-1 win over Torino on Tuesday.
"It wasn't easy against Catania at the weekend and it wasn't easy against Parma either. These are the sort of wins that win you the league," said Tevez, who will miss Sunday's trip to Napoli after being booked.
"I would like to have been there on Sunday. I don't like not playing."
Cristian Molinaro pulled a goal back for a spirited Parma just after the hour mark but they lost Amauri to a red card for apparently elbowing Giorgio Chiellini in an aerial challenge almost immediately afterwards.
"It gives me satisfaction to see how we played with 10 men, even if I think the red card was a bit excessive," said Parma boss Roberto Donadoni.
"I don't think we showed the same grit and determination in the first half that we showed in the second, but the reaction was good both in terms of determination and quality, which is encouraging."
Defeat for Parma left them sixth, level on 47 points with fifth-placed Inter Milan, who host Udinese on Thursday.
Atalanta are four points behind the pair after a 2-0 win over 18th-placed Livorno, their fifth victory in a row, thanks to goals from Giuseppe De Luca and German Denis.
Lazio slipped to eighth with a 2-0 loss at Genoa.
Antonio Conte's Juve moved up to 81 points and stayed 20 ahead of third-placed Napoli, who beat bottom side Catania 4-2 in a thrilling game in Sicily.
Rafael Benitez's team raced into a 4-0 halftime lead thanks to a double from Duvan Zapata, a tap-in from Jose Callejon following a comical defensive mix-up and a stunning volley from Henrique.
"We did very well in attack, but leading 4-0 at half-time made it difficult to maintain concentration," said Benitez.
"Our objective continues to be reaching Roma in second. We have to do our work, continue winning and see where we end up.
After the Juve game this weekend we'll have only one match per week and that will make it easier for us."
Napoli are now 10 points ahead of fourth-placed Fiorentina, having lost to Vincenzo Montella's side in Naples at the weekend, thanks to AC Milan, who relieved the pressure on their under-fire manager Clarence Seedorf with a 2-0 win in Florence.
Milan remain in 12th place on 39 points after a 23rd-minute header from Philippe Mexes and a delicate Mario Balotelli free kick just after the hour mark secured their first league win in five games, offering Seedorf a stay of execution after reportedly being given two matches to save his job.
"A lot of false things were said. I am open to criticism as long as it is constructive, but what I see here is the intention to create controversy by spreading lies disguised as fact," said a visibly angry Seedorf after the win.
"Those people who wrote certain things about me have to deal with their own conscience, because mine is clear.
"I tried to keep the players focused on the game, but it was impossible not to be aware of what was being said outside."
Cagliari were the big winners at the bottom of the table, taking advantage of Livorno's defeat by beating Hellas Verona 1-0 with a bullet Nene header and moving up to 32 points, eight clear of the drop zone.
They are five ahead of Chievo Verona, who inched three points away from from the bottom three with a thumping 3-0 win against Bologna.
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