  We classify locally homogeneous quasi-Sasakian manifolds in dimension five
that admit a parallel spinor $\psi$ of algebraic type $F \cdot \psi = 0$ with
respect to the unique connection $\nabla$ preserving the quasi-Sasakian
structure and with totally skew-symmetric torsion. We introduce a certain
conformal transformation of almost contact metric manifolds and discuss a link
between them and the dilation function in 5-dimensional string theory. We find
natural conditions implying conformal invariances of parallel spinors. We
present topological obstructions to the existence of parallel spinors in the
compact case.
