Walter Mazzarri's side twice trailed in the first half but a brace from Rodrigo Palacio either side of the break saw them draw level and then a 56th-minute strike from Fredy Guarin put the hosts ahead.
However, that lead only lasted three minutes before Parma striker Nicola Sansone made it 3-3 with his second of the match and, despite both sides having further chances in the final half-hour, the points where shared.
Parma's other goal was scored by Marco Parolo in first-half stoppage time following a terrible error by Inter keeper Samir Handanovic, giving the visitors a 2-1 lead at half-time.
The result leaves Inter four points adrift of Napoli and the final Champions League spot while Parma move up to 10th.
Mazzarri had demanded a dramatic improvement from his side following their midweek Coppa Italia scare against Trapani, when Inter sneaked a 3-2 win having led 3-0 at the break, but he must have been wondering if his words had fallen on deaf ears as Parma bossed the early exchanges.
They had already tested Handanovic with a Gabriel Paletta strike before taking an 11th-minute lead through Sansone, who finished emphatically after being put through on the edge of the box by a pinpoint Marco Marchionni pass.
Inter finally started to move up through the gears though and they created a number of chances to draw level, with Palacio missing the best opportunity when he fired over with the goal at his mercy.
The home side then survived a let-off in the 41st minute when Afriyie Acquah's effort was cleared off the goalline by Andrea Ranocchia, and within two minutes of that escape Inter were level.
Ranocchia drilled over a low cross from the right and Palacio got ahead of his marker to volley home from close range.
Inter's joy was short-lived though, as in first-half stoppage time Parma retook the lead following a glaring error by Handanovic.
The Slovenia international allowed Paletta's weak cross to slip through his legs as he dived forward, and Parolo had the easy task of sliding the ball into an empty net.
The goals continued to flow in the second half, and within 11 minutes of the re-start Inter found themselves ahead.
It was 2-2 in the 53rd minute when Palacio headed a fine left-wing cross from Ricardo Alvarez past Parma keeper Antonio Mirante, who got a hand to the ball but could only deflect it onto the inside of the post before seeing it rebound into the net, and then moments later, Guarin made it 3-2.
Palacio headed a Alvarez pass back to Guarin on the edge of the box and he turned inside a defender before drilling in a left-footed shot which took a decisive deflection off Parma defender Alessandro Lucarelli to leave Mirante wrong-footed as it rolled into the net.
Parma were not behind for long though as Sansone turned in Mattia Cassani's low cross to make it 3-3.
Both teams pushed for a winner after that, and Inter almost got it deep into stoppage-time when unmarked captain Esteban Cambiasso powered a volley goalward from eight yards out, but Mirante was able to parry it away as the points were shared.