Chances are the building is built with concrete block, and not actual poured concrete.
Cement stucco dosent work well in cold climates when applied directly over poured concrete.
Stucco is as a rule a cement base mortar and is applied to the block in several coats.
It has become fairly common to add decorative trim made from Styrofoam onto structures for added eye appeal.
The basic mortar product used for the concrete block, will not adhere properly to the Styrofoam, so a different product is used. There are a variety of brands, just as there are with paint, but a common name would be "Dryvit".
I doubt the woodpeckers are attacking the mortar based product on the main structure, that being the concrete block areas. But the Styrofoam trim areas could well have attracted something by way of insects which are causing it.
Those areas are also fairly easily damaged at ground level by landscape maintenance. Not having seen the building, this is of coarse an opinion of what might be. But assuming im correct, id be looking at pressure washing then coating the affected areas at least and possibly the entire building with a proper paint on product including those made by companies like Dryvit.