Should server spare parts be stored in climate controlled storage?

Assuming that the spares are in their original packaging or otherwise sealed then you shouldn't really need to do anything other than keep them dry. If they are sealed then they should be able to cope with a fairly humid atmosphere - though obviously the higher the humidity the more likely moisture is to creep into even the most tightly packaged item (unless it's airtight).

As long as the temperature range isn't unduly excessive (significantly below freezing or significantly above room temperature 21C) the temperature shouldn't be an issue as the parts aren't generating heat. In that respect a power supply is the same as a hard drive (for example).

The only thing to add here is that if the parts are kept somewhere cold they should be left to warm up to room temperature before fitting. This will avoid undue heat stress when they are first powered up.

Dust would only be an issue if the spares weren't kept in any packaging.

AlanBarber's comment is the best answer but in regard to whether or not there's a need to store the parts in a climate controlled environment you should consider where they've been prior to that.

Few factories are climate controlled. Even fewer warehouses are climate controlled. Even fewer still transport vehicles (trucks, planes, ships, etc.) are climate controlled. Those parts were created, stored and transported without the need or desire for climate control. Why should that change once they're in your possession?