  The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the
space of square integrable functions supported on a given parallelepiped in
$\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper,
we derived necessary and sufficient conditions for a parallelepiped in
$\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies
constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a
restriction relevant in many applications. In this paper, we investigate
analogous conditions for parallelepipeds that permit a Riesz basis of
exponentials with the same constraints on the frequencies. We provide a
sufficient condition on the parallelepiped for the Riesz basis case which
directly extends one of the necessary and sufficient conditions obtained in the
orthogonal basis case. We also provide a sufficient condition which constrains
the spectral norm of the matrix generating the parallelepiped, instead of
constraining the structure of the matrix.
