The 20-year-old Perea gave the holders the lead after 25 minutes of their last-16 encounter at the Stadio Olimpico.
However, Jonathan Biabiany cancelled out the hosts' opener just before the interval as Parma, who went into the tie unbeaten in nine games in all competitions, battled back.
Just two points separate the sides in the Serie A table, and it appeared that extra-time would be required to decide who progressed to the last eight.
But Colombian forward Parea had other ideas, as he struck with his second of the match at the death for the six-time Coppa Italia winners.
Lazio will now meet either Napoli or Atalanta in the quarter-finals.
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