It is essential to capture joint channel characteristics in cooperative multi-link systems in order to make realistic performance assessments and channel modeling. This paper investigates the joint statistical properties of large-scale parameters (LSPs) in high-speed railway (HSR) multi-link propagation scenarios. A dedicated long-term evolution (LTE) network-based multi-link channel sounding system is established, which breaks the constraints in HSR measurements and supports simultaneous time–frequency–space measurements for multiple links. To guarantee the effectiveness of multi-link measurements, a time-delay-window-based partition method is used to identify channel impulse responses from different links. Based on the processed multi-link data, the statistics of LSPs, involving shadow fading, K-factor, delay spread, and azimuth spread, is estimated and compared for the two links in a plain viaduct scenario on HSR. Then, auto-correlations and two kinds of cross correlations, such as intra-link correlation and inter-link correlation in the measured scenario, are calculated and analyzed. In addition, a simulation model of joint statistical characteristics for LSPs is proposed and validated according to the obtained measurement results. These results can be used to assess the actual performance of cooperative multi-link systems on HSR, and the proposed model can provide realistic inputs for channel models in HSR multi-link propagation scenarios.