"How is it that exam papers get leaked in this country? How is this right?" added Raj, 16.
Protesters were led by Abhijeet Dipke, a 30-year-old Boston University graduate who arrived in New Delhi from the United States on Saturday.
His parody party -- "Cockroach Janta Party" (CJP), a play on Prime Minister Narendra Modi's Hindu-nationalist Bharatiya Janata Party (BJP) -- has won millions of followers on social media since its launch last month.
"The youth of the country will no longer fear anyone, they will fight," Dipke, a former political communications strategist for the opposition Aam Aadmi Party, told supporters at the rally.
"Cockroaches don't ever fear, they never die either," said Dipke, as others shouted in unison.
The movement emerged after India's Chief Justice Surya Kant reportedly likened young people who criticised the government to "cockroaches" and "parasites" during a court hearing, sparking outrage among the youth. Kant later said his comments were taken out of context.
CJP's popularity has soared, using the slogan "a political front for the youth, by the youth, for the youth".