Juve were poor in the first half of a quarter-final they were expected to win comfortably against Serie A's bottom side.
They improved marginally after the break and secured their progression to the last four thanks to Alvaro Morata's 89th-minute goal, which gave them a 1-0 triumph.
It was far from Juve's best performance but Allegri said that was largely down to the way Parma played and the conditions.
"You can't win every time 2-0 or 3-0," he told RAI Sport.
"We played a good game and Parma gave us a hard time. There were difficulties, partly due to the pitch, and Parma played well.
"I think it's an important result because it shows that those who haven't played as much are ready and can be called upon at any time.
"We knew it wouldn't be easy and the side was mature not to allow Parma anything."
Goalkeeper Marco Storari deputised for veteran Gianluigi Buffon on his 37th birthday and he felt that Juve - aiming for a fourth successive Serie A crown this season - took Parma lightly.
"We started out a bit sluggish, I don't know if it's because we underestimated Parma," he said.
"Let's give credit to Parma, who worked for 90 minutes and put in a great performance.
"We have mental strength, even though we found a lot of difficulties tonight."
