  Extending the notion of parallelism we introduce the concept of approximate
parallelism in normed spaces and then substantially restrict ourselves to the
setting of Hilbert space operators endowed with the operator norm. We present
several characterizations of the exact and approximate operator parallelism in
the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a
Hilbert space $\mathscr{H}$. Among other things, we investigate the
relationship between approximate parallelism and norm of inner derivations on
$\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a
$C^*$-algebra by using states. Finally we utilize the linking algebra to give
some equivalence assertions regarding parallel elements in a Hilbert
$C^*$-module.
