Why did the left-wing political parties of Europe move toward center around 1983? How was the change rationalized at the time?

I can't provide an answer for left-wing parties in general but can provide an answer for the PCI (Italian Communist Party).

In terms of the PCI the move to 'Eurocommunism' was the culmination of years of a widening schism with the Soviet Union which had begun arguably in 1956 with the suppression of the Hungarian Revolution.

As the history of the PCI was the focus of both my undergraduate and MA dissertations, I could spend weeks discussing the minutiae of the PCI and how it went from being essentially the Italian expression of Stalinism to entering into a political agreement with the Christian Democrats. So will try and give the very condensed version.

Between 1928 and 1964 (bar a four year gap between 1934 and 1938), the party was led by the staunchly Stalinist leader Palmiro Togliatti. In 1956, following Khrushchev's scathing condemnaton of Stalinism he seemingly changed his views, although I seriously doubt he genuinely did. In any case, he stood by Khrushchev's decision to invade Hungary and crush the Hungarian Revolution of 1956. This had disastrous effects on the PCI as its electoral growth ground to a halt at the 1958 general election and some serious rifts started to open within the party. Famously, the leader of the Communist trade union CGIL, Giuseppe di Vittorio, openly criticised the leadership and denounced the Soviet invasion. To give you an idea of how much of a big deal this guy was, he was the the president of the World Trade Union Congress between 1949 and his death in 1957; on that topic, his funeral allegedly drew a crowd of million people against the 'only' million people at Togliatti's funeral.

Eventually, when Togliatti died in 1964, a close friend and ally (and former Stalinist), Luigi Longo, was elected as General Secretary. However, the left-wing block within the party managed to change quite signficantly the path of the PCI; for context, the left-wing block was the more progressive wing of the party and one which had been paying attention at the development of the New Left.

This more progressive outlook, meant that in 1968 the PCI broke ranks and condmened the Soviet invasion of Czechoslovakia. Meanwhile, internally the party was once again in turmoil as the more traditionalist 'old guard' had to contend with a quite serious challenge from the extraparliamentary left and from the more progressive elements within the party. So when Luigi Longo died in 1972, Enrico Berlinguer was elected as General Secretary; although one of Togliatti's favourite proteges, Berlinguer had shown very early on that he was unwilling to tow the Soviet party line.

Berlinguer was arguably one of the architects of Eurocommunism but he reached a position after years of questioning the Soviet Union and the relationship beween the PCI and the CPSU. The relative success of his position, which culminated with a definitive split with the Soviet Union was rationalised in a very simple way. In his view, the Soviet project had been exhausted and he believed that the dogmatic approach of the Soviet model in Eastern Europe had meant progress past a certain point was no longer possible. Therefore, it was necessary for Communist parties in the West to find their own way to achieve their goals which would take into consideration the very different circumstances in each country. In a televised interview, Berlinguer went as far as stating that although many of Marx's and Lenin's teachings were still important, they were no longer relevant to the world in 1982 and that Communists in the West ought to modify their perspective on how to achieve their goals and, to an extent, even modify their goals where appropriate.

I suspect that this is probably true of other left-wing parties in Europe but others will have to confirm (or correct this). This is also a highly distilled version of why the PCI went from being an Italian expression of Stalinism to Eurocommunism but more than happy to expand further.