In this paper, a bilevel multiobjective programming problem, where the lower level is a convex parameter multiobjective program, is concerned. Using the KKT optimality conditions of the lower level problem, this kind of problem is transformed into an equivalent one-level nonsmooth multiobjective optimization problem. Then, a sequence of smooth multiobjective problems that progressively approximate the nonsmooth multiobjective problem is introduced. It is shown that the Pareto optimal solutions (stationary points) of the approximate problems converge to a Pareto optimal solution (stationary point) of the original bilevel multiobjective programming problem. Numerical results showing the viability of the smoothing approach are reported.