No side has won the competition more times than Juve, but the Italian giants have not tasted success since 1995.
They seem destined for a fourth successive Scudetto crown, though, and a league and cup double would be an outstanding way for Massimiliano Allegri to cap his first season in charge.
His side will need to show more in the final third if they are to win the Coppa Italia for a 10th time, but Morata's late strike proved enough to seal their progression into the last four on this occasion.
The Spanish forward - introduced in the 77th minute - latched on to Fernando Llorente's ball before sliding his shot under Parma goalkeeper Antonio Mirante in smart fashion.
Simone Pepe saw his cross hit the post in the first half and luck looked to be against Juve when Claudio Marchisio had an effort deflected onto the crossbar in the 86th minute.
But Morata's late show ensured Allegri - who made eight changes - did not have to worry about extra time and, potentially, penalties.
Veteran midfielder Andrea Pirlo was back in the team for Juve, after recovering from flu, while first-choice goalkeeper Gianluigi Buffon took a spot on the bench on his 37th birthday.
The clash looked a mismatch on paper, with title-chasing defending champions Juve facing a Parma side who sit bottom of Serie A.
But the class divide was not evident during a poor and uninspiring first half, in which both sides lacked any sort of cutting edge.
Raffaele Palladino cut inside and shot just over in the 23rd minute for Parma, who were dealt a blow when defender Pedro Mendes pulled up injured in pursuit of 18-year-old Frenchman Kingsley Coman - leading to the introduction of Fabiano Santacroce.
The hosts built some pressure as half-time approached, but the best they had to show for it was an Antonio Nocerino effort that was turned behind by Giorgio Chiellini.
Juve nearly caught them cold when Pepe's cross from the left was allowed to bounce and hit the post with Mirante beaten in first-half stoppage time.
Arturo Vidal fired the rebound just over but the match could, seemingly, only improve after the break.
That was not the case, though, as Juventus battled to break down Roberto Donadoni's strugglers.
Chiellini's thumping header went over the crossbar from a Pirlo corner in the 55th minute, while Gabriel Paletta did the same for Parma at the other end.
The match was crying out for a spark and with 19 minutes remaining, Allegri threw Paul Pogba on looking for just that.
Morata was then introduced and he at least forced a save from Mirante after a mazy run, before Marchisio was denied by the crossbar.
As the visitors stormed forward, they did leave themselves open at the back - but that meant little in the end as Morata booked Juve's spot in the semi-finals against either Roma or Fiorentina with a cool finish.
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