This article presents a performance assessment of an electrical power and cooling system powered by a parabolic dish collector and using refrigerants with low global warming potential. The study was conducted using energy and mass balances for each component and system. The simulation includes various parameters, such as solar radiation, the focal temperature of the solar collector, the ambient temperature, the power cycle pressure ratio, and the chiller’s evaporation temperature. The results show that the efficiency of the organic Rankine cycle with the refrigerant R1233zd(E) is similar to that of the refrigerants R123 and R245fa and is up to 11 and 50 times lower than with R290 and R744, respectively. The solar absorption chiller using the refrigerant R717 can achieve cooling with a supply temperature up to 5 °C lower than that of R718. The dynamic simulation results show that the energy efficiency of the proposed solar-powered generator–chiller is 14% higher than that of a standard solar-powered absorption chiller. Furthermore, the same solar-powered generator–chiller reduces the primary energy required by a conventional system by 60% (PESr = 0.60). The presented results may be useful for the design of sustainable generator–chillers for rural areas or for autonomous housing in tropical climates.