Degradation observations of modern engineering systems, such as manufacturing systems, turbine engines, and high-speed trains, often demonstrate various patterns of time-varying degradation rates. General degradation process models are mainly introduced for constant degradation rates, which cannot be used for time-varying situations. Moreover, the issue of sparse degradation observations and the problem of evolving degradation observations both are practical challenges for the degradation analysis of modern engineering systems. In this paper, parametric inverse Gaussian process models are proposed to model degradation processes with constant, monotonic, and S-shaped degradation rates, where physical meaning of model parameters for time-varying degradation rates is highlighted. Random effects are incorporated into the degradation process models to model the unit-to-unit variability within product population. A general Bayesian framework is extended to deal with the degradation analysis of sparse degradation observations and evolving observations. An illustrative example derived from the reliability analysis of a heavy-duty machine tool's spindle system is presented, which is characterized as the degradation analysis of sparse degradation observations and evolving observations under time-varying degradation rates.