Brilliantly taken goals either side of half-time from Italian internationals Federico Dimarco and Nicolo Barella and Marcus Thuram's neat volley in the 67th minute were enough for the reigning champions to close in on Napoli.
Simone Inzaghi's Inter are also level on 31 points with second-placed Atalanta, who could claim the provisional league lead with a win over AC Milan later on Friday.
Napoli host Lazio on Sunday night after being dumped out of the Italian Cup by their capital city opponents on Thursday.
"We scored some great goals today... The only disappointing thing is that we conceded. We didn't deserve to given how we played over the 94 minutes," said Inzaghi to DAZN.
Inter are unbeaten in 14 matches in all competitions and travel to Leverkusen on Tuesday knowing that a win would almost certainly ensure direct qualification to the last 16 with two matches remaining.
Parma, who netted through a late Matteo Darmian own goal, sit 12th in Serie A, but Inzaghi still picked his best available XI even with a big European fixture around the corner, selecting captain and star striker Lautaro Martinez for his 300th Inter appearance.
Martinez has struggled for form this season and as well as having a first half strike rightly ruled out for offside the Argentina forward also wasted three good opportunities to score.
The 27-year-old had his head in his hands when he poked wide a great chance one-on-one with Parma goalkeeper Zion Suzuki, six minutes after Thuram lashed home his 10th league goal of the season.
France attacker Thuram was born in Parma where his father Lilian played for five seasons, winning the UEFA Cup and Italian Cup in 1999 as part of a star-studded outfit which also featured Italian icons Gianluigi Buffon and Fabio Cannavaro.
