In the parallel calibration for transmitting phased arrays, the calibration receiver must separate the signals belonging to different antenna elements to avoid mutual interference. Existing algorithms encode different antenna elements' radiation with orthogonal signature codes, but these algorithms are far from desired for large-scale spaceborne antenna arrays. Considering the strictly limited resources on satellites, to improve hardware efficiency of large-scale spaceborne antenna arrays, in this work inspired by the idea of non-orthogonal multiple access (NOMA) we design a series of non-orthogonal signature codes for different antenna elements by Cyclically Shifting an m-Sequence (CSmS) with different offsets named as CSmS-NOMA signaling. This design can strike an elegant balance between the performance and complexity and is very suitable for large-scale spaceborne antenna arrays. It is shown that no matter how many antenna elements there are to be calibrated simultaneously, CSmS-NOMA signaling needs only one calibrating waveform generator and one matched filter. Hence it is much more efficient than the existing fully orthogonal schemes. In order to evaluate the achievable calibration accuracy, a unified theoretical framework is developed based on which the relationship between calibration accuracy and signal to noise ratio (SNR) has been clearly revealed. Furthermore, a hardware experiment platform is also built to assess the theoretical work. For all the considered scenarios, it can be concluded that the theoretical, simulated and experimental results coincide with each other perfectly.