Medical officials said more than 30 people - including both protesters and police - had been injured.
Later, Moldovan state TV said one woman had "choked to death from carbon monoxide poisoning in the parliament".
Although election observers concluded that last weekend's vote was fair, opposition parties and many students are not convinced.
The Communist Party won 50% of votes in the election. They were followed by the Liberal Party with almost 13% of the votes, the Liberal Democratic Party with 12%, and Our Moldova Alliance on almost 10%.
It is still not clear whether the Communists will win the 61 seats in the 101-seat parliament that they need to elect Mr Voronin's successor unopposed.
President Voronin is barred by the constitution from running for a third term, although he has indicated he wants to remain involved in affairs of state.
If no president is chosen before 8 June, another parliamentary election must be held.
The pro-Western centre-right opposition parties have said they will not join a coalition with the Communists, who favour strong links with both Russia and the European Union.
Mr Voronin's successor will lead the poorest country in Europe, where the average wage is just under $250 (£168) a month, and will inherit an unresolved conflict over the breakaway region of Trans-Dniester.
The region has run its own affairs, with Moscow's support, since the end of hostilities in a brief war in 1992. Mr Voronin resumed direct talks with Trans-Dniester last year.
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