  Utilizing the Birkhoff--James orthogonality, we present some
characterizations of the norm-parallelism for elements of
$\mathbb{B}(\mathscr{H})$ defined on a finite dimensional Hilbert space,
elements of a Hilbert $C^*$-module over the $C^*$-algebra of compact operators
and elements of an arbitrary $C^*$-algebra. We also consider the
characterization of norm parallelism problem for operators on a finite
dimensional Hilbert space when the operator norm is replaced by the Schatten
$p$-norm. Some applications and generalizations are discussed for certain
elements of a Hilbert $C^*$-module.
