  We study the concepts of Birkhoff--James orthogonality and parallelism in
Hilbert space operators, induced by the operator radius norm $w_{\rho}(\cdot)$.
In particular, we completely characterize Birkhoff--James orthogonality and
parallelism with respect to $w_{\rho}(\cdot)$. As an application of the results
presented, we obtain a well-known characterization due to R.~Bhatia and
P.~\v{S}emrl for the classical Birkhoff--James orthogonality of Hilbert space
operators. Some other related results are also discussed.
