The former TV star won 73% of the vote in last month's election - and has already outlined controversial plans for a snap election.
Since last month's election, Zelenskiy's opponents in the Rada sought to put off the inauguration close to the May 27 deadline by which the parliament can be dissolved.
Zelenskiy's victory reflected Ukrainians' exhaustion with widespread corruption and the country's political elite.
He ditched the idea of a traditional motorcade to his inauguration, walking to parliament in the capital of Kiev through a park packed with people. Beaming, he stopped and high-fived supporters, even at one point jumping up to kiss a man on the top of his head.
Zelensky, a comedian who previously played the president of Ukraine in a television series, had no real-life political experience prior to his presidential run.
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Poroshenko led Ukraine for five years, overseeing the fallout over Russia's annexation of Crimea and armed conflict with Moscow-backed separatists in the country's East that has caused some 13,000 deaths.
"To all Ukrainians, no matter where you are, I promise that I will never let you down", Zelensky had said after the results were announced. "It wasn't us who started that war. And for that, we need a parliament that functions".
As ministers and lawmakers listened with dismay, Zelensky urged everyone in the cabinet to resign, asking them to "free the spot for people who will think about the future generations, not about the future elections".
Two of the three officials he has asked to leave, Defence Minister Stepan Poltorak and SBU chief Vasyl Grytsak, promptly handed in their resignations.
Zelenskiy and his supporters are eager to hold elections sooner hoping he will benefit from his current support and to allow him to begin passing his own legislative agenda.
Zelenskiy called on lawmakers to adopt bills that would strip them of immunity and ban illegal enrichment. Political factions have been maneuvering for weeks trying to avoid such a situation.
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New Ukrainian President Volodymyr Zelenskiy took his oath of office and delivered an inaugural speech in which he announced the dissolution of parliament in Kiev.
A representative of the Ukrainian Election Commission told the Ukrainian media that if a snap election is called, no court ruling can cancel it.
Parliamentary rules prohibit it from being disbanded for 30 days after a coalition breaks up, a delay that would create another delay since it would mean the parliament would already be in the final six months of its a term, when it can not be dissolved.
In his speech, Zelensky referred to his background as a comedian.
The 41-year-old comedian said his first task is to end the conflict with Moscow-backed separatists in the country's east, but added that dialogue with Russian Federation could only happen after the return of Ukrainian territory and prisoners of war.
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