I'm not sure whether the procession towards parliamentary system happened because of the EU. I don't know much about other countries, but in the Finnish case, the move towards parliamentary system was very much a response to uncertainties about division of responsibilities, excesses and abuses of power that became prevalent during president Urho Kekkonen's 25-year reign. Kekkonen was first elected in 1956 and proceeded to dominate Finnish politics until he was essentially forced to step down due to health issues in 1982, at 82 years of age.

There is still endless debate about the motivations, reasons and outcomes of Kekkonen's reign, with opinions ranging from "great patriot who saved Finland from becoming Soviet satellite" to "ruthless egomaniac who couldn't let go of power" to "unwitting or even willing agent of KGB". The truth is likely to be rather complex and there are elements of all three in his story, with the "egomaniac who couldn't let go of power" aspect decidedly shadowing his later years. (He also believed he'd live longer if he kept on working.)

What Kekkonen is best remembered for is his willingness to ruthlessly use the "Eastern" or "Moscow card" to beat internal opponents. Increasingly over the years, he thought he and he alone could stave off the Soviet encroachment (he apparently believed the Soviet Union was going to win the Cold War) with his personal diplomacy, and was very quick to brand anyone holding contrary opinions as a danger to Finland's foreign relations (that is, angering the Soviet Union). On its part, the Soviet Union was determined to keep Kekkonen in power, and helped him with concessions and trade deals and the like at crucial moments - almost certainly also by instigating a so-called 1961 note crisis that caused a major scare in Finland and, conveniently, caused the main opponent of Kekkonen's to bow out from the presidential race "for foreign policy reasons."

For various reasons, many related to relentless browbeating by Kekkonen or his supporters, most other political parties more or less acquiesced or had been shunted to eternal opposition, partly because the president held so much power in the Finnish system (and unofficially, even more). Kekkonen was prone to dissolving parliaments that did not please him, and things got so bad that in 1973 his term was extended by an "emergency" law and in 1978 elections, no serious opponents dared to challenge him. Kekkonen went on to win 259 of the 300 electoral college votes even though he didn't bother to appear in televised debates or otherwise campaign unduly. A common joke from those days says Finland's name should be changed to Kekkoslovakia, and that wasn't too far from the truth.

After he was forced out of power, major constitutional overhauls were initiated to ensure that similar escapades would not happen in future. These included term limitations, restricting the role of president in cabinet formation, mandating the approval of Prime Minister before dissolving the Parliament, and making the Prime Minister more responsible for foreign policy, which had been the sole prerogative of the president. These changes brought about somewhat confusing situations when it wasn't fully clear who actually was in charge of Finland's EU policy for example, but examples of these break the 20-year rule, and suffice to say that this confusion led to further cutbacks in president's role.

The end result is that Finland today has mostly parliamentary system instead of strong presidency, but EU had only an incidental role in all of this.

For sources, Suomen historian pikkujättiläinen provides an overview, as do most English-language contemporary histories of Finland.