Information-theoretic secrecy is studied for a parallel relay (diamond) network, in which a transmitter wishes to communicate to a receiver through two relay nodes. While there is no direct link between the transmitter and receiver and all flow of information has to be transmitted through the relays, the message has to be kept secret from each of them. The exact secrecy capacity is characterized for the network under the linear deterministic model. The problem is then studied when each terminal is equipped with multiple antennas, and the channels are parallel Gaussian links. Lower and upper bounds for the secrecy capacity are derived, and the gap is bounded by a constant independent of the channel parameters and SNR. This results in an approximate characterization for the secrecy capacity of the parallel Gaussian diamond network.