The theory and techniques of compressed sensing (CS) have shown their potential as a breakthrough in accelerating k-space data acquisition for parallel magnetic resonance imaging (pMRI). However, the performance of CS reconstruction models in pMRI has not been fully maximized, and CS recovery guarantees for pMRI are largely absent. To improve reconstruction accuracy from parsimonious amounts of k-space data while maintaining flexibility, a new CS SENSitivity Encoding (SENSE) pMRI reconstruction framework promoting joint sparsity (JS) across channels (JS CS SENSE) is proposed in this paper. The recovery guarantee derived for the proposed JS CS SENSE model is demonstrated to be better than that of the conventional CS SENSE model and similar to that of the coil-by-coil CS model. The flexibility of the new model is better than the coil-by-coil CS model and the same as that of CS SENSE. For fast image reconstruction and fair comparisons, all the introduced CS-based constrained optimization problems are solved with split Bregman, variable splitting, and combined-variable splitting techniques. For the JS CS SENSE model in particular, these techniques lead to an efficient algorithm. Numerical experiments show that the reconstruction accuracy is significantly improved by JS CS SENSE compared with the conventional CS SENSE. In addition, an accurate residual-JS regularized sensitivity estimation model is also proposed and extended to calibration-less (CaL) JS CS SENSE. Numerical results show that CaL JS CS SENSE outperforms other state-of-the-art CS-based calibration-less methods in particular for reconstructing non-piecewise constant images.