Is pith a ground tissue with no specialized function?
Short answer: it's not the greatest question.

Long Answer:

So there are three primary tissues in a plant: ground tissue, dermal tissue, and vascular tissue. See below for how they sort out:



However, some sources refer to the tissues of plants slightly differently, and indicate that the three basic tissue types are parenchyma, collenchyma, and schlerenchyma. 


Example: "Botany: A Lab Manual" by Pfluger 2014:


  The three fundemental tissues of plants are parenchyma, collenchyma, and sclerenchyma



Since these are all "ground" tissues, this approach to classifying plant tissues would suggest that all plant parts consist of some kind of ground tissue (including all the plant parts you mention in your post). In fact, this is often the case...


Again, from Pfluger 2014:


  They [parenchyma] can be found in virtually any part of the plant and are the most common cell type in a plant. 



Of the parts you mention, xylem is "specialized" to move water through the plant via tracheids, pits, and vessel elements; phloem moves sugars/sap through sieve tube cells which are supported by companion cells; cambium acts in secondary growth of wood and bark via the vascular and cork cambia; and bark is a water-proof protective layer. 

However, "pith" is simply the central parenchyma cells in the stem, and therefore are primarily just used for storage like "all unspecialized" ground tissue.  

As for cambium: from here:


  Interfascicular cambium differentiates from parenchyma or collenchyma cells located between the vascular bundles (mainly in stem)