The protests are led by former international cricket star turned opposition leader Imran Khan and cleric Tahir ul-Qadri, who runs a network of Islamic schools and charities.
Legislator Marvi Memon said lawmakers would discuss the political crisis in today's parliamentary session.
Exhausted protesters, some carrying blankets or colourful umbrellas, were resting on the grass on Constitution Avenue, the main street housing government buildings, on Wednesday after some used bolt cutters to force their way past barricades of barbed wire and shipping containers late on Tuesday.
Riot police did not intervene.
They want Sharif to stand down over allegations of corruption and rigging last year's polls.
The protests have raised questions over the stability of nuclear-armed Pakistan. The young democracy of 180 million people has a history of coups and the government is struggling to tackle high unemployment, daily power cuts and a Taliban insurgency.
"Now no police nor army will stop us," Khan told supporters on Tuesday. "Nawaz Sharif, resign by tomorrow 8 p.m., otherwise we will come to the prime minister's house."
The country's powerful military, which often acts as an arbiter when it is not ruling directly, has called for a political solution to the crisis.
"Situation requires patience, wisdom and sagacity from all stakeholders to resolve prevailing impasse through meaningful dialogue in larger national and public interest," military spokesman General Asim Bajwa tweeted as the protesters approached parliament.
Parliament, the Supreme Court, the prime minister's office and the entrance to the enclave where Western embassies are located are all on the avenue taken over by protesters.
Khan wants Sharif to step down because he believes the prime minister rigged last year's polls. Sharif won the election by a landslide, and the polls were the first democratic transfer of power in Pakistan's history.
Qadri wants Sharif to step down because he says the system is corrupt. He has promised free housing for the homeless, and welfare and subsidised food and electricity for the poor.