This paper presents two fast algorithms for total variation-based image reconstruction in partially parallel magnetic resonance imaging (PPI) where the inversion matrix is large and ill-conditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproblems. The first method reduces the image reconstruction problem to an unconstrained minimization problem, which is solved by an alternating proximal minimization algorithm. One phase of the algorithm solves a total variation (TV) denoising problem, and second phase solves an ill-conditioned linear system. Linear and sublinear convergence results are given, and an implementation based on a primal-dual hybrid gradient (PDHG) scheme for the TV problem and a Barzilai-Borwein scheme for the linear inversion is proposed. The second algorithm exploits the special structure of the PPI reconstruction problem by decomposing it into one subproblem involving Fourier transforms and another subproblem that can be treated by the PDHG scheme. Numerical results and comparisons with recently developed methods indicate the efficiency of the proposed algorithms.