We present a computational model of the mechanics of growth of the trophoblast bilayer in a chorionic villous, the basic structure of the placenta. The placental trophoblast is modeled as a collection of elastic neutrally buoyant membranes (mononuclear cytotrophoblasts and multinucleated syncytiotrophoblast) filled with a viscous, incompressible fluid (cytoplasm) with sources of growth located inside cells. We show how this complex, dynamic fluid-based structure can be modeled successfully using the immersed boundary method. The results of our research presented here include simulations of two processes-cell proliferation and cell fusion which both play a crucial role in the growth and development of the trophoblast tissue. We present the computed results of simulations of both processes running independently as well as simultaneously, along with comparisons with clinically obtained results.