The main problem with the hardware implementation of turbo codes is the lack of parallelism in the MAP-based decoding algorithm. This paper proposes to overcome this problem with a new family of turbo codes, called Slice Turbo Codes. This family is based on two ideas: the encoding of each dimension with P independent tail-biting codes and a constrained interleaver structure that allows parallel decoding of the P independent codewords in each dimension. The optimization of the interleaver is described. A high degree of parallelism is obtained with equivalent or better performance than the best known turbo codes. The parallel architecture allows reduced complexity turbo decoding for very high throughput applications.