We study the performance of policy gradient methods for the subclass of Markov games known as Markov potential games (MPGs), which extends the notion of normal-form potential games to the stateful setting and includes the important special case of the fully cooperative setting where the agents share an identical reward function. Our focus in this paper is to study the convergence of the policy gradient method for solving MPGs under softmax policy parameterization, both tabular and parameterized with general function approximators such as neural networks. We first show the asymptotic convergence of this method to a Nash equilibrium of MPGs for tabular softmax policies. Second, we derive the finite-time performance of the policy gradient in two settings: 1) using the log-barrier regularization, and 2) using the natural policy gradient under the best-response dynamics (NPG-BR). Finally, extending the notion of price of anarchy (POA) and smoothness in normal-form games, we introduce the POA for MPGs and provide a POA bound for NPG-BR. To our knowledge, this is the first POA bound for solving MPGs. To support our theoretical results, we empirically compare the convergence rates and POA of policy gradient variants for both tabular and neural softmax policies.