Hybrid radio frequency (RF)/free space optical (FSO) systems are among the candidate enabling technologies for the next generation of wireless networks since they benefit from both the high data rates of the FSO subsystem and the high reliability of the RF subsystem. In this paper, we focus on the problem of throughput maximization in the parallel hybrid RF/FSO relay channel. In the parallel hybrid RF/FSO relay channel, a source node sends its data to a destination node with the help of multiple relay nodes. Thereby, for a given relay, the source-relay and the relay-destination FSO links are orthogonal with respect to each other due to the narrow beam employed for FSO transmission, whereas, due to the broadcast nature of the RF channel, half-duplex operation is required for the RF links if self-interference is to be avoided. Moreover, we consider the two cases where the relays are and are not equipped with buffers. For both cases, we derive the optimal relay selection policies for the RF and FSO links and the optimal time allocation policy for transmission and reception for the RF links. The proposed optimal protocols provide important insights for optimal system design. Since the optimal buffer-aided (BA) policy introduces an unbounded end-to-end delay, we also propose a suboptimal BA policy which ensures certain target average delays. Moreover, we present distributed implementations for both proposed optimal protocols. Simulation results demonstrate that a considerable gain can be achieved by the proposed adaptive protocols in comparison with benchmark schemes from the literature.