  In this work we show that if the frame property of a Gabor frame with window
in Feichtinger's algebra and a fixed lattice only depends on the parity of the
window, then the lattice can be replaced by any other lattice of the same
density without losing the frame property. As a byproduct we derive a
generalization of a result of Lyubarskii and Nes, who could show that any Gabor
system consisting of an odd window function from Feichtinger's algebra and any
separable lattice of density $\frac{n+1}{n}$, $n \in \mathbb{N}_+$, cannot be a
Gabor frame for the Hilbert space of square-integrable functions on the real
line. We extend this result by removing the assumption that the lattice has to
be separable. This is achieved by exploiting the interplay between the
symplectic and the metaplectic group.
