Physics-informed neural networks (PINNs) are widely used to solve forward and inverse problems in fluid mechanics. However, the current PINNs framework faces notable challenges when presented with problems that involve large spatiotemporal domains or high Reynolds numbers, leading to hyper-parameter tuning difficulties and excessively long training times. To overcome these issues and enhance PINNs' efficacy in solving inverse problems, this paper proposes a spatiotemporal parallel physics-informed neural networks (STPINNs) framework that can be deployed simultaneously to multi-central processing units. The STPINNs framework is specially designed for the inverse problems of fluid mechanics by utilizing an overlapping domain decomposition strategy and incorporating Reynolds-averaged Navier-Stokes equations, with eddy viscosity in the output layer of neural networks. The performance of the proposed STPINNs is evaluated on three turbulent cases: the wake flow of a two-dimensional cylinder, homogeneous isotropic decaying turbulence, and the average wake flow of a three-dimensional cylinder. All three turbulent flow cases are successfully reconstructed with sparse observations. The quantitative results along with strong and weak scaling analyses demonstrate that STPINNs can accurately and efficiently solve turbulent flows with comparatively high Reynolds numbers.