Martinez replaced Joao Mario in the 77th minute and made his mark within two minutes.
Inter won possession in midfield and Radja Nainggolan slipped the ball to the Argentine who held off his marker and fired his shot into the roof of the net.
Luciano Spalletti's side remained third in Serie A, and on course for the Champions League next season, with 43 points while Parma are 12th with 29.
"It's an important win which gives us strength and enthusiasm," said Inter coach Luciano Spalletti, who has been under increasing pressure following Champions League and Coppa Italia exits in the last two months.
"We have the potential to overcome our difficulties, although in some moments it seems that we’re weighed down by not winning and we drop below our usual standard."
The visitors, whose previous three league games produced a 0-0 draw with Sassuolo and 1-0 defeats to Torino and lowly Bologna, had a letoff before halftime when former Liverpool and Roma forward Gervinho crashed a shot against the Inter bar.
Danilo D'Ambrosio thought he had broken Inter's duck when he turned the ball in from a corner in the 54th minute but it was chalked off after a VAR review showed that the ball struck his arm.
Inter gained in confidence after the goal and Matias Vecino saw a shot cleared off the line while Marcelo Brozovic hit the upright with a low shot.
"Lautaro has fighting spirit and he does really well when the ball comes to him," Spalletti said. "We showed positive things tonight, but we need to produce further performances like this to gain consistency."
(Writing by Brian Homewood, editing by Pritha Sarkar and Ed Osmond)
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