  Using Cohen's classification of symplectic reflection groups, we prove that
the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic
reflection group are themselves symplectic reflection groups. This is the
symplectic analogue of Steinberg's Theorem for complex reflection groups. Using
computational results required in the proof, we show the non-existence of
symplectic resolutions for symplectic quotient singularities corresponding to
three exceptional symplectic reflection groups, thus reducing further the
number of cases for which the existence question remains open. Another
immediate consequence of our result is that the singular locus of the
symplectic quotient singularity associated to a symplectic reflection group is
pure of codimension two.
