  We consider Dolbeault-Dirac operators on quantized irreducible flag manifolds
as defined by Kr\"ahmer and Tucker-Simmons. We show that, in general, these
operators do not satisfy a formula of Parthasarathy-type. This is a consequence
of two results that we prove here: we always have quadratic commutation
relations for the relevant quantum root vectors, up to terms in the quantized
Levi factor; there are examples of quantum Clifford algebras where the
commutation relations are not of quadratic-constant type.
