But Blaise Matuidi secured the three points just before the hour mark - the France World Cup victor latching onto a fine Mandzukic back-heel before firing high into the net from a tight angle.
But he is yet to open his account for his new club, and Allegri believes even a player as great as Ronaldo will need time to adapt to Italian football.
Parma almost took the lead shortly before the break as Gervinho turned provider to pick out Emiliano Rigoni, whose low strike was palmed away by Wojciech Szczesny. "The worldwide break will make him sharper".
Parma should have been level in the ninth minute when Gervinho played Antonio Di Gaudio through on goal, but the striker failed to score with the net gaping.
Ronaldo's hunt for his first competitive goal for Juventus continues when the Italian giants host Sassuolo in Serie A on September 16.
Ronaldo said it was "the best goal" of his career and it helped Real to a 3-0 win on the night and moved them a step closer to an eventual quarter-final victory over Juve.
The Old Lady's held on to claim a precious 2-1 win with the pair of Mario Mandzukic and Blaise Matuidi getting the needed goals.
The forward's best chance came when a loose ball landed at his feet but Ronaldo shot wide and was also left frustrated when Parma goalkeeper Luigi Sepe managed to intercept a cross and took the ball off his head.
The Portuguese has not scored in his first three Serie A outings, but his coach was full of praise for his new star player.