The aim of this article is to show that solvers for tridiagonal Toeplitz systems of linear equations can be efficiently implemented for a variety of modern GPU-accelerated and multicore architectures using OpenACC. We consider two parallel algorithms for solving such systems with special assumptions about coefficient matrices. As the first algorithm, we propose a new, faster implementation of the divide and conquer method.The next algorithm is a new, vectorizable algorithm based on a recently introduced sequential method. We consider the use of both column-wise and row-wise storage formats for two-dimensional arrays and show how to efficiently convert between these two formats using cache memory and improve the overall performance of our implementations. We also show how to tune the performance by predicting the best values of the methods' parameters. Numerical experiments performed on Intel CPUs and Nvidia GPUs show that our new implementations achieve relatively good performance and accuracy.