We propose deep learning algorithms for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process is derived by a Brownian motion and an independent compensated Poisson random measure. In the novel algorithm for coupled FBSDEJs, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on the deep FBSDE method. For FBSDEJs with small-to-moderate jump sizes and moderate jump intensities we propose the novel FBSJNN framework in which a single neural network to approximate the PIDE solution is used, while leveraging Taylor expansion to eliminate the need for a separate approximation of the non-local integral term. For both the deep learning algorithms, we derive the error estimates by exploring the error bound of Euler time discretization and the simulation error of deep learning algorithm. For the former, it is also shown that the approximation error converges to zero given the universal approximation capability of neural networks. several numerical examples are provided to show the efficiency of these proposed algorithms.