This paper proposes a novel alternative to the Logarithmic Bahl-Cocke-Jelinek-Raviv (Log-BCJR) algorithm for turbo decoding, yielding significantly improved processing throughput and latency. While the Log-BCJR processes turbo-encoded bits in a serial forwards-backwards manner, the proposed algorithm operates in a fully-parallel manner, processing all bits in both components of the turbo code at the same time. The proposed algorithm is compatible with all turbo codes, including those of the LTE and WiMAX standards. These standardized codes employ odd-even interleavers, facilitating a novel technique for reducing the complexity of the proposed algorithm by 50%. More specifically, odd-even interleavers allow the proposed algorithm to alternate between processing the odd-indexed bits of the first component code at the same time as the even-indexed bits of the second component, and vice-versa. Furthermore, the proposed fully-parallel algorithm is shown to converge to the same error correction performance as the state-of-the-art turbo decoding algorithm. Owing to its significantly increased parallelism, the proposed algorithm facilitates throughputs and latencies that are up to 6.86 times superior to those of the state-of-the art algorithm, when employed for the LTE and WiMAX turbo codes. However, this is achieved at the cost of a moderately increased computational complexity and resource requirement.