In 1926, H. Levy proved that a second order symmetric parallel non singular tensor on a space of constant curvature is a constant multiple of the metric tensor. R. Sharma generalized Levy's result and studied second order parallel tensor on a Kähler space of constant holomorphic sectional curvature as well as on a compact K-contact manifold without boundary. U.C. De and Debasish Tarafdar showed that for a P-Sasakian manifold M with a symmetric parallel tensor of type (0,2) is also a constant multiple of the metric tensor and there is no parallel 2-form on M.

In this paper, We have studied symmetric and skew-symmetric second order parallel tensor on a Kenmotsu manifold.