  This paper is concerned with the construction of new solutions in terms of
birational maps to the functional tetrahedron equation and parametric
tetrahedron equation. We present a method for constructing solutions to the
parametric tetrahedron equation via Darboux transformations. In particular, we
study matrix refactorisation problems for Darboux transformations associated
with the nonlinear Schr\"odinger (NLS) and the derivative nonlinear
Schr\"odinger (DNLS) equation, and we construct novel nine-dimensional
tetrahedron maps. We show that the latter can be restricted to six-dimensional
parametric tetrahedron maps on invariant leaves. Finally, we construct
parametric tetrahedron maps employing degenerated Darboux transformations of
NLS and DNLS type.
