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.
But Morata's late show ensured Allegri - who made eight changes - did not have to worry about extra time and, potentially, penalties.
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.
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.
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.
