Absolutely parallel grammars are defined and it is shown that the family of languages generated is equal to the family of languages generated by two-way deterministic finite-state transducers. Furthermore it is shown that this family forms a full AFL [4], is properly contained in the family of languages generated by two-way nondeterministic finite-state transducers (which is equal to the family of checking automata languages [8]) and properly contains the family of nonexpansive context-free languages [7].