A resolution overwhelmingly supported by the assembly linked Mr Yanukovich, who was ousted on Saturday and is now on the run, to police violence against protesters which it said had led to the deaths of more than 100 citizens from Ukraine and other states.
The Hague-based court said it would need a request from the government of Ukraine giving it jurisdiction over the deaths.
With early elections set for May 25, one of Ukraine's most prominent opposition figures, retired world boxing champion Vitaly Klitschko, confirmed he would run for president.
Mr Yanukovich was indicted for mass murder on Monday over the shooting of demonstrators and is now on the wanted list, having last been seen at Balaclava in Crimea, near Russia's Sevastopol naval base.
An aide said be on the run with Yanukovich was shot in the leg, his spokesman said yesterday. It was not clear where the aide, Andriy Klyuev, was, or whether he with the fugitive leader.
The parliament said former interior minister Vitaly Zakharchenko and former prosecutor-general Viktor Pshonka, who are also being sought by the authorities, should also be sent for trial at the ICC.
Authorities under Mr Yanukovich had systematically abused their power, the resolution said, with police torturing protesters and holding activists naked in freezing temperatures.
The Hague-based International Criminal Court, which since its founding in 2002 has handled only cases from Africa, said it could intervene if Ukraine requested it to.
Mr Yanukovich's fall has revived fears Ukraine might split along the fault line that divides its pro-Western and pro-Russian regions.
In a fresh warning to the EU and US not to try to shape Ukraine's future, Russian Foreign Minister Sergei Lavrov said Ukraine must not be forced to choose between Russia and the West.
Both Russia and the West, while competing for influence over Ukraine under its new rulers, have said publicly that they do not want a split to happen.
Moscow has said it will not deal with those who led an "armed mutiny" against Mr Yanukovich, who was backed by Russia, and said it fears for the lives of its citizens. Many Russians live in the Russian-speaking east and Crimea.
Mr Lavrov said: "It is dangerous and counter-productive to try to force upon Ukraine a choice on the principle: 'You are either with us or against us',"
Both Russia and the West should use political contacts in Ukraine to calm the situation down and not seek advantage at a time when national dialogue is needed, Mr Lavrov added.
Unrest erupted in Ukraine after Mr Yanukovich abandoned a proposed trade pact with the EU and turned instead towards Moscow, which offered loans and cheaper supplies of gas.
The EU's foreign policy chief said Russia should behave like a good neighbour and let Ukraine move forward in the way it chooses after three months of conflict.
Catherine Ashton, the first senior foreign official to visit Kiev since the overthrow of Mr Yanukovich, said the EU understood the need for strong links between Kiev and Moscow but that a strong message should be sent about Ukraine's territorial integrity. Voicing "strong support" for Ukraine's new leaders, Baroness Ashton urged them to form an "inclusive" government and focus on getting the country through short-term problems.
She said the EU would work with the International Monetary Fund but the IMF would make its own assessment of the situation.
The Finance M inistry in Kiev has said the country needs £21 billion in foreign financial assistance over the next two years and says the money needs to start coming through in the next week or two.
European Commission President Jose Manuel Barroso said he was committed to supporting Ukraine.