Both teams had chances to score in the first half, with Inter goalkeeper Samir Handanovic forced into a fine save by the hosts' Jasmin Kurtic, while Inter's Romelu Lukaku could not convert from a good position just before the interval.
Antonio Conte's side showed their quality after the break as Sanchez latched on to a loose ball to score in the 54th minute and the Chilean forward added a second eight minutes later after brilliant work from Lukaku.
Struggling Parma stayed 19th, six points from safety, and without a league win since November.
"I'm like a lion -- the more I play, the better I feel," Sanchez told Sky Sport Italia.
"We have to fight to win the Scudetto. We have to think about ourselves and not to make mistakes. The players have grown a lot in the last few months."
Sanchez, the former Barcelona, Arsenal and Manchester United forward, was given just his ninth league start of the season by coach Conte and first since the beginning of February, with Inter's second-top scorer Lautaro Martinez on the bench.
Having scored his first Inter goal since November at the weekend, Sanchez sneaked his first over the line before slotting home his fifth Serie A strike of the season to put his side in command.
Hernani's goal put the visitors under pressure, with Parma piling forward, but Inter defended in numbers and saw out the win.
Milan's surprise draw against Udinese on Wednesday opened the door for Inter to widen the gap at the top, with the celebrations at the final whistle showing just how important a victory this could be in the title race.
We are in a very good position, we are ahead of everyone," Conte said. "There has been an important improvement but there are 13 games left and the Italian championship is a battle.
"We cannot drop points, we have 13 finals to play."