Is there a reason why paratrooper units in the Cold War era and beyond are usually the ones stirring lots of controversy? In France, it was the Paras who threatened to seize Paris and Corsica if de Gaulle wasn’t installed as President. In the UK, there was Bloody Sunday.
In the case of the Algerian War, it wasn't simply paratroopers who were involved in the crisis of 1958, but rather the regular French army as well as the French Foreign Legion. It's a complicated sequence of events which is hard to summarize and that deals with matters of Cold War radicalization of the regular elements of the French Armed Forces.I've written a previous answer on this topic here, but it is at best a general and brief overview.