In this paper, we prove that if there exists a second order symmetric parallel tensor on an almost Kenmotsu manifold $(M^{2n+1},\phi,\xi,\eta,g)$ whose characteristic vector field $\xi$ belongs to the $(k,\mu)'$-nullity distribution, then either $M^{2n+1}$ is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold, or the second order parallel tensor is a constant multiple of the associated metric tensor of $M^{2n+1}$. Furthermore, some properties of an almost Kenmotsu manifold admitting a second order parallel tensor with $\xi$ belonging to the $(k,\mu)$-nullity distribution are also obtained.