Dual-function radar communication (DFRC) systems implement both sensing and communication using the same hardware. Such schemes are typically more efficient in terms of size, power, and cost, over using distinct radar and communication systems. Since these functionalities share resources such as spectrum, power, and antennas, DFRC methods typically entail some degradation in both radar and communication performance. In this work we propose a multi-carrier agile phased array radar (MAPAR) DFRC, which extends the previously proposed frequency agile radar and multi-carrier agile radar to a phased array antenna operating with constant modulus waveforms. The inherent spatial and spectral randomness of the proposed MAPAR is utilized to convey digital messages in the form of index modulation. The resulting communication scheme naturally coexists with the radar functionality, and thus does not come at the cost of reduced radar performance. We carry out a theoretical analysis of the radar and communication performances, quantifying the radar beam pattern and the achievable bit rate. Our numerical results demonstrate that the proposed MAPAR yields an effective range-spatial beam pattern, similar to that achievable with traditional phased array radar, while its communications bit rate is comparable to utilizing an independent communication module.