  Let $\mathcal{H}$ be a Hilbert space, and let $K(\mathcal{H})$ be the
$C^*$-algebra of compact operators on $\mathcal{H}$. In this paper, we present
some characterizations of the norm-parallelism for elements of a Hilbert
$K(\mathcal{H})$-module by employing the Birkhoff--James orthogonality. Among
other things, we present a characterization of transitive relation of the
norm-parallelism for elements in a certain Hilbert $K(\mathcal{H})$-module. We
also give some characterizations of the Schatten $p$-norms and the operator
norm-parallelism for matrices.
