Anyone knows what are the conditions like in Cordillera Blanca in January / February? Apparently it's the rainy season but has anyone climbed there during these months?

I've been to Peru during the rainy season (in the drier parts of the country), but not the Blanca.

Even if you hit a clear day, avalanche danger would be extreme on many of the routes there.

If you must go to Peru then, I'd head for the drier regions (such as around Arequipa). You can still have bad weather then, but it would make a much more reasonable destination in the rainy season.

Outside Peru, there are plenty of places in season in South America during that time period (i.e. Colombia, much of Ecuador, Chile, Argentina).

See this thread as well; the same basic comments would apply, except that the bugs would be out in full force:
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