Why was Strasbourg designated the official seat of the European Parliament? 

A 1992 decision formalised a situation that already existed at the time and which reflected compromises arrived at over a number of years.

When the European Coal and Steel Community (ECSC) was set up a few years after WWII, in 1952, establishing joint management of the steel and coal reserves of six countries, including Germany and France, its institutions were located in Luxembourg. The Council of Europe (an intergovernmental body made up of 47 countries championing human rights and culture was also set up in the immediate post-WW2 period), was already based in Strasbourg and it offered its plenary chamber for meetings of the ECSC's "Common Assembly", which was to develop into the European Parliament. Strasbourg gradually became the main home of plenary sessions of the Parliament, though additional sessions were also held in Luxembourg in the 1960s and 1970s.

After the creation of the European Economic Community in 1958, much of the work done by the European Commission and the Council of Ministers came to be concentrated in Brussels. Since Parliament's work involves closely monitoring and interacting with both these institutions, over time Members decided to organise more of their work in Brussels. By the early nineties, the present arrangement was more or less in place, with committees and political groups meeting in Brussels and the main plenary sessions taking place in Strasbourg. A major part of Parliament's staff is based in Luxembourg.