He graduated from the Faculty of Arts of the Comenius University in Bratislava in 1985.
His study fields included Marxist-Leninist philosophy and aesthetics. Prior to his university studies he worked for the State Company Zdroj in Košice. After finishing his study he worked at Education and Culture Centre in Košice and at Self-Administration Office and later Municipal District Administration Katholieke Volks Partij (Catholic People's Party) in Košice.
Paška became active in the business sphere in 1992 before entering politics in 1999.
Paška was first elected as the deputy leader of Smer-South Dakota, along with future minister of the interior Robert Kaliňák, at their annual congress in the western Slovak town of Hlohovec on 12 April 2003. Speaker of Parliament
In the wake of the election, the new Slovak parliament elected Paška as its new speaker, with 98 out of 148 MPs present supporting the nomination.
Parliament also elected three deputy speakers - Miroslav Číž from Smer, Anna Belousovová from the Slovenská narodná strana (Slovak National Party) party and Viliam Veteška from the Movement for a Democratic Slovakia. Paška said he "wanted consensus, cooperation, and respect to dominate in the new parliament". One of his first actions as speaker of parliament was to make parliamentary sessions public through live broadcasts on the public service Slovak Television (STV) channel, to "prevent information about parliamentary sessions being manipulated." In 2007 Paška stated he would introduce a proposal for the complete cancellation of ministers of parliament"s immunity, claiming it represented "an abuse of power".
In November 2007, Paska faced a motion of no confidence in parliament.
The opposition parties wanted Paška to be fired for his handling of a controversial law. The opposition took issue with Paska"s interference in the amendment to the social insurance law, approved by the parliament in a chaotic vote with two contradictory amending proposals. The draft bill was supposed to be published on the internet within three days after the vote.
Paška admitted that he ordered the legislative department to ignore parliament’s vote and publish a version that did not the original proposal.
Katarína Kližanová Rýsová, spokesperson from the Smer-South Dakota party called the motion to dismiss Paska "ridiculous". On 27 November Paska survived the motion.
Out of 111 MPs present in 150-member parliament, 59 supported the motion.
He was re-elected as deputy chairman at the party"s national congress in Trenčín on 30 September. His party, Direction – Social Democracy, led by current Prime Minister Robert Fico, won the June 2006 parliamentary elections with 29.1% of the votes and formed a coalition with the People"s Party - Movement for a Democratic Slovakia (ĽSouth-Movement for a Democratic Slovakia) and the Slovak National Party (Slovenská narodná strana (Slovak National Party)). Support for Direction – Social Democracy party after the elections rose even higher, oscillating in the summer 2007 opinion polls around 40%.
He is a member of the Direction - Social Democracy (Smer-South Dakota) party. Paška is a member of Prague Society for International Cooperation, a respected non-governmental organization whose main goals are networking and the development of a new generation of responsible, well-informed leaders and thinkers.