  In this paper, we establish several characterizations of the $A$-parallelism
of bounded linear operators with respect to the seminorm induced by a positive
operator $A$ acting on a complex Hilbert space. Among other things, we
investigate the relationship between $A$-seminorm-parallelism and
$A$-Birkhoff-James orthogonality of $A$-bounded operators. In particular, we
characterize $A$-bounded operators which satisfy the $A$-Daugavet equation. In
addition, we relate the $A$-Birkhoff-James orthogonality of operators and
distance formulas and we give an explicit formula of the center mass for
$A$-bounded operators. Some other related results are also discussed.
