In this work, we perform Bayesian inference tasks for the chemical master equation in the tensor-train format. The tensor-train approximation has been proven to be very efficient in representing high-dimensional data arising from the explicit representation of the chemical master equation solution. An additional advantage of representing the probability mass function in the tensor-train format is that parametric dependency can be easily incorporated by introducing a tensor product basis expansion in the parameter space. Time is treated as an additional dimension of the tensor and a linear system is derived to solve the chemical master equation in time. We exemplify the tensor-train method by performing inference tasks such as smoothing and parameter inference using the tensor-train framework. A very high compression ratio is observed for storing the probability mass function of the solution. Since all linear algebra operations are performed in the tensor-train format, a significant reduction in the computational time is observed as well.