Millimeter wave (mm-wave) communication with large-scale antenna array configuration is seen as the key enabler of the next generation communication systems. Accurate knowledge of the mm-wave propagation channels is fundamental and essential. In this contribution, a novel complexity-efficient high resolution parameter estimation (HRPE) algorithm is proposed for the mm-wave channel with large-scale uniform circular array (UCA) applied. The proposed algorithm is able to obtain the high-resolution estimation results of the spherical channel propagation parameters. The prior channel information in the delay domain, i.e., the delay trajectories of individual propagation paths observed across the array elements, is exploited, by combining the high-resolution estimation principle and the phase mode excitation technique. Fast initializations, effective interference cancellations and reduced searching spaces achieved by the proposed schemes significantly decrease the algorithm complexity. Furthermore, the channel spatial non-stationarity in path gain across the array elements is considered for the first time in the literature for propagation parameter estimation, which is beneficial to obtain more realistic results as well as to decrease the complexity. A mm-wave measurement campaign at the frequency band of 28-30 GHz using a large-scale UCA is exploited to demonstrate and validate the proposed HRPE algorithm.