Given some multidimensional simplex and a subsimplex of possibly lower dimension, the subsimplex will be called axially parallel if each of its edges is parallel to some edge of the base simplex. This class of subsimplices is intimately connected to some multivariate generalizations of convexity in two ways: First these convexity properties are determined by the behavior of the function on parallel subsimplices, and second, they are preserved by restriction on a subsimplex if and, with some minor restrictions, only if the subsimplex is parallel.