A hypersurface \(M\) of a Riemannian manifold \(A\) is said to be cyclic- parallel, if its shape operator \(A\) satisfies \({\mathfrak S} \langle (\nabla_ X A)Y, Z\rangle = 0\) for all \(X\), \(Y\), \(Z\) tangent to \(M\) where \(\mathfrak S\) denotes the cyclic sum with respect to \(X\), \(Y\), \(Z\). In the paper the following is proved: Let \(N\) be the quaternionic projective space \(QP^ m\) \((m \geq 2)\). Then \(M\) is cyclic-parallel if and only if \(M\) is congruent to an open subset of a tube around a canonically embedded \(QP^ k\) with \(k \in \{0, \dots, m-1\}\). A crucial step in the proof is \textit{J. Berndt's} classification of curvature-adapted real hypersurfaces in \(QP^ m\) [J. Reine Angew. Math. 419, 9-26 (1991; Zbl 0718.53017)].