Parametric problems in the transient analysis refer to the investigations of the relationship between parameters and dynamic behavior. To solve parametric problems, we propose a Galerkin method-based polynomial approximation. The proposed method uses a series of polynomials combined with coefficients to approximately represent state variables with parameters, where the Galerkin method is introduced to determine the coefficients. Thereby, the relationship between parameters and dynamic behavior is established through the polynomial expressions. Because models in power system transient analysis are a set of complicated differential-algebraic equations, practical issues related to the non-polynomial model terms, parameterization of initial operation point are discussed in this paper. The proposed method is tested on a three-machine system and the IEEE 145-bus test system. The case studies show that compared to trajectory sensitivities, the proposed method can maintain high accuracy in cases of large parameter variations and strong non-linearity. We also illustrate that the validity of the proposed method for parametric problems like trajectory approximation, finding the critical parameter values and uncertainty quantification.