  In this article, we first consider the $L^{2}$ \textit{Morse-Novikov
cohomology} on a complete Riemannian manifold $M$ equipped with a parallel
$1$-form which includes Vaisman manifold. Based on a vanishing theorem of
$L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic
forms on $M$ are identically zero.
