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In the existing cellular networks, it remains a challenging problem to communicate with and control an unmanned aerial vehicle (UAV) swarm with both high reliability and low latency. Due to the UAV swarm's high working altitude and strong ground-to-air channels, it is generally exposed to multiple ground base stations (GBSs), while the GBSs that are serving ground users (occupied GBSs) can generate strong interference to the UAV swarm. To tackle this issue, we propose a novel two-phase transmission protocol by exploiting cellular plus device-to-device (D2D) communication for the UAV swarm. In Phase I, one swarm head is chosen for ground-to-air channel estimation, and all the GBSs that are not serving ground users (available GBSs) transmit a common control message to the UAV swarm simultaneously, using the same cellular frequency band, to combat the strong interference from occupied GBSs. In Phase II, all the UAVs that have decoded the common control message in Phase I further relay it to the other UAVs in the swarm via D2D communication, by exploiting the less interfered D2D frequency band and the proximity among UAVs. In this paper, we aim to characterize the reliability performance of the above two-phase protocol, i.e., the expected percentage of UAVs in the swarm that can decode the common control message, which is a non-trivial problem due to the complex system setup and the intricate coupling between the two phases. Nevertheless, we manage to obtain an approximated expression of the reliability performance of interest, under reasonable assumptions and with the aid of the Pearson distributions. Numerical results validate the accuracy of our analytical results and show the effectiveness of our protocol over other benchmark protocols. We also study the effect of key system parameters on the reliability performance, to reveal useful insights on the practical system design.