The Catalans beat Deportivo Alaves 3-1 on May 27 to secure a third triumph in the competition in as many seasons and give departing coach Luis Enrique a victorious send-off.
However, the club declined to hold a celebratory parade amid a lingering sense of disappointment surrounding the 2016-17 campaign.
They lost the Liga crown to Real Madrid and watched their bitter rivals from the capital become the first team to win back-to-back Champions League titles since the tournament's rebranding in 1992, with Zinedine Zidane's side beating Juventus 4-1 in the final.
Comparisons have been drawn between Zidane's Madrid and Pep Guardiola's Barcelona, who won 14 trophies between 2008 and 2012 in a period of dominance in Spain and Europe.
Pique, however, claimed the teams will remain incomparable until Barca hold a bus parade to celebrate a solitary Copa triumph in three years – something Madrid did in 2011.
"You still can't compare what we have won during all these years with what Madrid have won in the last two seasons," Pique told Movistar +.
"What we did made Real Madrid go on a victory parade just for winning the Copa del Rey.
"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."
Pique was mocked by Madrid fans during the players' celebration of their league title triumph in the capital, with players including Sergio Ramos, Isco and Dani Carvajal visibly amused by chants of "Pique, b******, salute the champions".
The Spain international, who has often been accused of antagonising Madrid with comments on social media, insisted he was "not offended at all" by the incident and did not want an apology.