The two-dimensional melting of a binary mixture of cell tissues is numerically investigated in the presence of rigidity disparity. We present the full melting phase diagrams of the system by using the Voronoi-based cellular model. It is found that the enhancement of rigidity disparity can induce a solid-liquid transition at both zero temperature and finite temperature. (i) In the case of zero temperature, the system undergoes a continuous solid-hexatic transition followed by a continuous hexatic-liquid transition for zero rigidity disparity, but a discontinuous hexatic-liquid transition for finite rigidity disparity. Remarkably, the solid-hexatic transitions always arise when the soft cells reach the rigidity transition point of monodisperse systems. (ii) In the case of finite temperature, the melting occurs via a continuous solid-hexatic transition followed by a discontinuous hexatic-liquid transition. Our study may contribute to the understanding of solid-liquid transitions in binary mixture systems with rigidity disparity.