We analyze an open system comprised of two parallel totally asymmetric simple exclusion processes with particle attachment and detachment in the bulk under partially asymmetric coupling conditions. The phase diagrams are obtained using boundary layer analysis of continuum mean-field equations and characterized for different values of lane-changing rates. The structure of the phase diagram remains qualitatively the same as the one in fully asymmetric coupling conditions up to a certain critical order of lane-changing rates, after which significant changes are found in the phase diagram. The effect of system size on the steady-state dynamics has also been examined. To validate theoretical findings, extensive Monte Carlo simulations are carried out.