Police fired tear gas and charged at protesters with batons on Monday after demonstrators tried to breach security barricades surrounding the protest site in central New Delhi, marking one of the most serious confrontations in the movement's campaign against Prime Minister Narendra Modi's government.
Despite the crackdown, many protesters continued marching towards parliament in small groups.
Cockroach Janta Party spokesperson Saurav Das accused authorities of "brutally cracking down" on "peaceful protesters".
Delhi Police said the march had not been authorised and restrictions were needed to maintain law and order around parliament during the monsoon session, which began on Monday.
The crowd at Jantar Mantar, a designated site for protests just a few kilometres from parliament, included students, professionals and families with young children.
As the day progressed, the gathering spilled beyond the protest site, with demonstrators filling surrounding streets and roads.
Many carried backpacks with water and food, expecting a long day. Others waved Indian flags as chants against the education minister and Modi rippled through the growing crowd.
The Cockroach Janta Party emerged in May, after the Supreme Court chief justice compared some unemployed young people to "cockroaches" during an unrelated hearing.
Supporters embraced the remark, turning it into a satirical movement that has attracted millions of followers on social media.
The movement has gained momentum after repeated leaks of high-stakes entrance exams for medical colleges and government jobs sparked public anger.
It drew wider attention after police forcibly moved hunger-striking activist Sonam Wangchuk to the hospital on Saturday.
Protesters calling themselves "cockroaches" have camped at Jantar Mantar for the past month, with members of several student groups joining Wangchuk's hunger strike.
Thousands of supporters have also turned out at universities and rallies in other cities.
They are demanding the resignation of Education Minister Dharmendra Pradhan, reforms to the examination system and compensation for the families of students who died by suicide following paper leaks.
Senior leaders in Modi's government have taken a hard line against the movement.
Pradhan has accused the protesters of acting against the nation, while other ministers have acknowledged students' concerns but insisted there was no need for talks or negotiations with the movement.
Though that appeared to shift on Monday, when a movement representative said protest leaders held talks with Health Minister JP Nadda, in one of the first apparent signs of government outreach to protest leaders.
The government did not immediately confirm the meeting, which came as protesters sought to bring their demands directly to MPs on the opening day of parliament's monsoon session, prompting authorities to tighten security around the complex.
Separately, activist Wangchuk appealed for permission to temporarily leave the hospital and join the march.
In a handwritten note, he also said he would end his nearly three-week hunger strike only if the government took responsibility for failures in the education system and exam paper leaks, or if protesters were allowed to reach parliament and MPs pledged to raise the issues in both houses.
The protests have become a rare public challenge to Modi's Hindu nationalist government in its third term, drawing support from opposition parties, rights activists and some Bollywood celebrities.
Protesters say space for dissent has shrunk under Modi, whose government has been criticised for suppressing public protests by jailing activists and, in some cases, using forceful police crackdowns.
