In this paper, we examine strong structural controllability of a particular subspace of linear time-invariant networks, namely, the null space of the parameterized family of system matrices. In this direction, we establish a one-to-one correspondence between the set of input nodes for null space controllability and the notion of skew zero forcing sets. Using this class of zero forcing sets, we provide conditions to guarantee that the null space of the parameterized set of state matrices sharing a common network topology is trivial. Moreover, the uncontrollability of the zero mode of directed and undirected networks from a single node is discussed. In addition, methods for growing a network while preserving strong structural controllability of its null space from a set of control nodes is presented. Finally, we provide an application of the developed results for the bipartite consensus dynamics.