A new type of grammar, called a parallel leveled grammar, is introduced. The families of languages generated by such grammars with contextfree, linear or right-linear subrules are studied. Right-linear parallel finite-leveled languages can be displayed as nested vector expressions, which are extensions of regular expressions. Various hierarchy theorems for these families of languages are obtained. A normal form for context-free parallel leveled grammars is derived and connections of these with other families of languages are investigated.