Classically one has results of

*L. P. Eisenhart* [Trans. Am. Math. Soc. 25, 297-306 (1923)] that a Riemannian manifold admitting a second order symmetric parallel tensor other than a constant multiple of the metric is reducible and

*H. Levy* [Ann. Math. 27 (1926)] that a second order symmetric parallel tensor on a space of constant curvature is a constant multiple of the metric. In [Int. J. Math. Math. Sci. 12, No. 4, 787-790 (1989;

Zbl 0696.53012)] the author considered parallel tensor fields that are not necessarily symmetric for both Riemannian and Kähler manifolds. In the present paper the author studies parallel tensor fields on contact metric manifolds. The results of the paper are the following: Ona

$K$-contact manifold a second order symmetric parallel tensor is a constant multiple of the metric. On a Sasakian manifold there are no non-zero parallel 2-forms; this generalizes the result for compact Sasakian manifolds of

*S. I. Goldberg* and the reviewer [J. Differ. Geom. 1, 347-354 (1967;

Zbl 0163.439)]. As an application of the first result the author proves that an affine Killing vector field on a compact

$K$-contact manifold without boundary is Killing.