Individual perception of disaster risk is not only the product of individual factors, but also the product of social interactions. However, few studies have empirically explored the correlations between rural residents’ flat social networks, trust in pyramidal channels, and disaster-risk perceptions. Taking Sichuan Province—a typical disaster-prone province in China—as an example and using data from 327 rural households in mountainous areas threatened by multiple disasters, this paper measured the level of participants’ disaster-risk perception in the four dimensions of possibility, threat, self-efficacy, and response efficacy. Then, the ordinary least squares method was applied to probe the correlations between social networks, trust, and residents’ disaster-risk perception. The results revealed four main findings. (1) Compared with scores relating to comprehensive disaster-risk perception, participants had lower perception scores relating to possibility and threat, and higher perception scores relating to self-efficacy and response efficacy. (2) The carrier characteristics of their social networks significantly affected rural residents’ perceived levels of disaster risk, while the background characteristics did not. (3) Different dimensions of trust had distinct effects on rural residents’ disaster-risk perceptions. (4) Compared with social network variables, trust was more closely related to the perceived level of disaster risks, which was especially reflected in the impact on self-efficacy, response efficacy, and comprehensive perception. The findings of this study deepen understanding of the relationship between social networks, trust, and disaster-risk perceptions of rural residents in mountainous areas threatened by multiple disasters, providing enlightenment for building resilient disaster-prevention systems in the community.