We theoretically investigate the properties of neutral and charged excitons and of the biexciton in vertically coupled quantum dots, as a function of the in-plane magnetic field B-parallel to. The main effect of the field consists in the suppression of the bonding-antibonding splitting, and in the resulting enhancement of the interdot correlations. As a consequence, the excitons form with the additional carrier a bound or an unbound complex depending on the sign of the charging, whereas the biexciton undergoes a transition between different quantum states with increasing B-parallel to. The discussed behaviors and transitions show up in the field dependence of experimentally accessible quantities, such as the charged-exciton and biexciton binding energies.