The capacity of parallel intensity-modulation and direct-detection (IM-DD) optical wireless channels with total average intensity and per-channel peak intensity constraints is studied. The optimal intensity allocation at low signal-to-noise ratio (SNR) is derived, leading to the capacity-achieving onoff keying (OOK) distribution. Interestingly, while activating the strongest channel is optimal if (i) the peak intensity is fixed, this is not the case if (ii) the peak intensity is proportional to the average intensity. The minimum average optical intensity per bit is also studied, and is characterized for case (i) where it is achievable at low SNR. However, in case (ii), the average optical intensity per bit grows indefinitely as SNR decreases, indicating that lower optical intensity per bit can be achieved at moderate SNR than at low SNR.