Correspondents say the city hall and ruling party headquarters are also in flames.
MPs have suspended a vote on changing the constitution to allow Mr Compaore to stand for re-election next year.
Five people has been killed in the protests, one of the most serious against Mr Compaore's rule, reports BBC Afrique's Yacouba Ouedraogo from the capital.
The main opposition leader, Zephirin Diabre, has called on the military to side with "the people".
He first took power in a coup in 1987, and has won four disputed elections since then.
"October 30 is Burkina Faso's Black Spring, like the Arab Spring," opposition activist Emile Pargui Pare told AFP news agency.
About 1,500 people breached the security cordon at parliament, AFP reports.
Protesters were setting fire to documents and stealing computer equipment and cars outside the building were also set on fire, it reports.
• Took power after Sankara was killed in mysterious circumstances by a group of soldiers in 1987
• First elected president in 1991 and again in 1998
• A new constitution in 2000 limited presidents to two term limits in office and limited the term to five years
• Won two further terms
• Protests at attempts to amend the term limits began a year ago, fuelled by the high cost of living
A government helicopter flying overhead was firing tear gas at them, Reuters reports.
The government has been forced to suspend Thursday's parliamentary vote on a constitutional amendment that would have lifted the limit on presidential terms so that Mr Compaore could run for office again in 2015.
It is not clear whether the government intends to hold the vote at a later stage, correspondents say.
Both France and the European Union (EU) have called on him to scrap the proposed constitutional amendment.
The EU said it could jeopardise Burkina Faso's stability. The US has also raised concern about the proposed amendment.