In this paper, transmission over buffer-aided diamond relay systems under statistical quality of service (QoS) constraints is studied. The statistical QoS constraints are imposed as limitations on delay violation probabilities. In the absence of channel state information (CSI) at the transmitter, truncated hybrid automatic repeat request-incremental redundancy (HARQ-IR) is incorporated to make better use of the wireless channel and the resources for each communication link. The packets that cannot be successfully received upon the maximum number of transmissions will be removed from buffer, i.e., outage occurs. The \emph{outage effective capacity} of a communication link is defined as the maximum constant arrival rate to the source that can be supported by the \emph{goodput} departure processes, i.e., the departure that can be successfully received by the receiver. Then, the outage effective capacity for the buffer-aided diamond relay system is obtained for HARQ-IR incorporated transmission strategy under the \emph{end-to-end} delay constraints. In comparison with the DF protocol with perfect CSI at the transmitters, it is shown that HARQ-IR can achieve superior performance when the SNR levels at the relay are not so large or when the delay constraints are stringent.