Gervinho found a way through 19 seconds into the second half - the third straight game he has scored against Inter - and added a second shortly after the hour mark.
That had Inter facing the real prospect of back-to-back home league defeats, though substitute Marcelo Brozovic pulled one back 137 seconds later to give his side some hope.
Inter had a strong penalty shout rejected but Perisic glanced in a last-gasp equaliser to deny Parma, who can still hold claim to being unbeaten in five away games with Inter at San Siro.
45:19 - Gervinho scored after just 19 seconds from the start of the second half: it is the earliest goal for Parma in a single Serie A half of play since Lanzafame, against Livorno in May 2010 (45:14). Hissing.#InterParma pic.twitter.com/Mpve021c0H— OptaPaolo (@OptaPaolo) October 31, 2020
Perisic, used as a makeshift forward in the absence of the injured Romelu Lukaku, had wasted a good chance 99 seconds in when volleying over the bar from close range.
Samir Handanovic kept out Andreas Cornelius but it was Inter who were initially the brighter, Luigi Sepe denying Lautaro Martinez before Achraf Hakimi nodded over.
Those missed chances would come back to haunt the hosts when Gervinho guided home Hernani's deep pass with a crisp first-time finish soon after the interval.
Andrea Ranocchia then had a goal ruled out for offside, shortly before Roberto Inglese played a ball clean through for Gervinho to strike under the onrushing Handanovic.
Brozovic pulled one back almost immediately for Inter, however, curling past a wrong-footed Sepe just five minutes after being introduced from the bench.
Antonio Conte's side had a penalty appeal turned down when Botond Balogh brought down Perisic in the box, while Ranocchia was somehow thwarted by a point-blank Sepe save.
But the late pressure finally told as Perisic got his head to Aleksandar Kolarov's free-kick to steer it past a helpless Seppe, who was grateful to see Arturo Vidal send a header of his own wide in the fourth minute of added time.
What does it mean? Inter struggle again
Conte was unwilling to risk the long-term fitness of striker Lukaku, who sustained a thigh injury in the goalless draw with Shakhtar Donetsk, for what appeared to be a routine home fixture.
Already without the services of Alexis Sanchez, Inter lacked a cutting edge in the final third for large parts, converting only two of their 25 attempts.
They have taken five points from the last 12 on offer in Serie A and are winless in two Champions League matches ahead of Tuesday's trip to Real Madrid, with the pressure now rising on the Inter boss.
Gervinho stars at San Siro
The Parma forward set the tone for a gripping second half with his fine finish for the opening goal, watching Hernani's pass all the way and firing past Handanovic.
His second was also impressive, as he peeled off Stefan de Vrij and kept his composure. Gervinho scored from his only two shots and also laid on a key pass in an impressive personal display.
More needed from Martinez
Martinez was the focal point of Inter's attack in the absence of Lukaku, with Perisic joining him up top and the equally disappointing Christian Eriksen just behind the front two until he was substituted.
The Argentina international had three shots on target - more than any other player on the field - but could not find a way past Sepe as his scoreless run stretched to a fifth game.