Here we describe a parallel cold standby system of four units which operate in two phases. In phase-I, there are two identical units that work in parallel and two more units (called phase-II units) are taken as spare in cold standby mode. The units of phase-I are non-identical with that of the units of phase-II. In the second phase, the cold standby units become simultaneously operative at the failure of all the units of phase-I. The single unit of phase-II cannot work alone due to its less efficiency. The repair activities of the units of the both phases are tackled by a single server who attends the system immediately as per the requirement of the user. The provision of priority for the operation of the units of phase-I has been made. Some well-known reliability characteristics such as mean sojourn time (MST), mean time to system failure (MTSF), long run availability, expected number of repairs of the units of phase-I and Phase-II, server’s busy period and expected number of visits and finally the expression for the profit function are studied for arbitrary values of different transition rates between states. The behavior of these characteristics has been observed and shown graphically. The present study has the application in power distribution systems.