Parma stunned the San Siro minutes before half time when substitute midfielder Stanko Juric (26) curled a beautiful long-range shot into keeper Andre Onana's top-right corner.
Inter, who beat Juventus 4-2 after extra time to lift the trophy in May last year, controlled possession but failed to muster a single shot on target until striker Lautaro Martinez (25) levelled in the 88th minute.
The Argentine collected a rebound on the edge of the box and lashed the ball inside goalkeeper Gianluigi Buffon's (44) right post.
Defender Francesco Acerbi (34) grabbed the winner for Inter with a header five minutes into the second half of extra time, having collected the ball from Buffon, who parried an earlier attempt by Federico Dimarco (25).
Parma, who beat Salernitana and Bari to reach the last 16, saw their goalkeeper Buffon return from a lengthy thigh injury to make his 50th, and likely last career appearance at the San Siro, 25 years after making his competition debut.
Inter's local rivals AC Milan host Torino in their last-16 clash on Wednesday, while Jose Mourinho's AS Roma take on Genoa on Thursday. Fiorentina play Sampdoria earlier that day.
