The Colombian's double ensured the Biancocelesti sealed their place in the last eight and will now meet either Napoli or Atalanta in the next round
The 20-year-old Perea gave the Biancocelesti 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 Perea 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.