Both of you are partially correct- the far-left is discredited by the war in the same way Nazism was discredited by World War II- sure, outright fascism was frowned upon, but the right wing in general survived.
Here it would be mostly the same- nobody in their right mind would identify as a Communist in the west, and certainly not as a hardline Stalinist (we're talking about breaking-the-lizardman-constant levels of unpopularity here), but DemSocs and certainly SocDems will live to see another day.
That said, people who now identify as the above but previously identified as actual Communists are very much fucked- for instance, Berlinguer renames the PCI to the PSD (Partido Socialista Democratico) but still doesn't gain anywhere near as much traction as he did OTL (although that also makes him less of a threat, and he lives another decade).
Regardless, plenty of left-wing parties are doing the purging of extreme elements themselves, as Labour ended up doing before Crosland brought them back into Number 10.