Plastic deformation of sphere-forming triblock thermoplastic elastomers is studied with molecular dynamics simulations in order to elucidate microscopic mechanisms operative in nanostructured macromolecular materials. Phase separation of linear triblock copolymers is achieved by first equilibrating a melt using soft interactions that are subsequently replaced by a standard bead spring model to obtain mechanical properties. We compare two deformation modes of both uniaxial stress and strain and vary the polymer chain length from unentangled to moderately entangled chains. Our simulations show that triblocks exhibit significant increase of strain hardening compared to homopolymer elastomers. We analyze several properties such as global chain deformation and local monomer motion, number and shape of spherical domains, and the evolution of the fraction of chains bridging between domains. Results confirm the notion of improved mechanical properties through effectively cross-linking chain ends. Void nucleation at different stages of deformation is observed to occur either at the interface between glassy and rubbery phases or immediately following the breakup of glassy domains and is therefore intimately related to the elastic heterogeneity of the material.