Barcelona beat Alaves 3-1 in the Copa del Rey final last month to end Luis Enrique's disappointing final season at the Nou Camp on a high.
But the Catalan giants refused to celebrate their victory with an open-top bus parade, unlike Real Madrid did in 2011 after their solitary Copa triumph in three years.
Real won a second successive Champions League crown and La Liga title this season under the guidance of manager Zinedine Zidane.
And the Frenchman's side have drawn comparisons to Pep Guardiola's Barcelona, who won 14 trophies between 2008 and 2012.
Gerard Pique takes dig at rivals Real Madrid
Gerard Pique aims dig at Barcelona rivals Real Madrid
But Pique insists Barcelona's dominance in Spain and Europe will remain unrivalled until they decide to hold a celebratory parade for a cup win.
And in a swipe at Real, Pique told Movistar: "You still can't compare what we have won during all these years with what Madrid have won in the last two seasons.
"What we did made Real Madrid go on a victory parade just for winning the Copa del Rey.
Real Madrid have won back-to-back Champions League crowns under Zinedine Zidane Real Madrid celebrate winning La Liga Sun, May 21, 2017
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Real Madrid have been crowned champions of La Liga
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"The day you see us going on a victory parade for winning the Copa will be the day when you can say that Madrid have entered a dominant cycle."
Real Madrid players appeared to take a swipe at Pique themselves after their 4-1 win over Champions League final win over Juventus on Saturday.
Sergio Ramos, Alvaro Morata and Kiko Casilla cut the net from one of the goals as a memento – something Pique is famous for doing.