  In this paper we study the classifying theory of principal bundles in the
parametrized setting, motivated by recent interest in higher gauge theory.
Using simplicial techniques, we construct a product-preserving classifying
space functor for groups in the category of spaces over a fixed space B.
Additionally, we prove that the fiberwise geometric realization functor sends a
large class of simplicial parametrized principal bundles to ordinary
parametrized principal bundles. As an application we show that the fiberwise
geometric realization of the universal simplicial principal bundle for a
simplicial group G in the category of spaces over B gives rise to a
parametrized principal bundle with structure group |G|.
