  In this article we consider systems of parallel hard {\it superellipsoids},
which can be viewed as a possible interpolation between ellipsoids of
revolution and cylinders. Superellipsoids are characterized by an aspect ratio
and an exponent $\alpha$ (shape parameter) which takes care of the geometry,
with $\alpha=1$ corresponding to ellipsoids of revolution, while
$\alpha=\infty$ is the limit of cylinders. It is well known that, while hard
parallel cylinders exhibit nematic, smectic, and solid phases, hard parallel
ellipsoids do not stabilize the smectic phase, the nematic phase transforming
directly into a solid as density is increased. We use computer simulation to
find evidence that for $\alpha\ge\alpha_c$, where $\alpha_c$ is a critical
value which the simulations estimate to be in the interval 1.2--1.3, the
smectic phase is stabilized. This is surprisingly close to the ellipsoidal
case. In addition, we use a density-functional approach, based on the
Parsons--Lee approximation, to describe smectic and columnar ordering. In
combination with a free--volume theory for the crystalline phase, a theoretical
phase diagram is predicted. While some qualitative features, such as the
enhancement of smectic stability for increasing $\alpha$, and the probable
absence of a stable columnar phase, are correct, the precise location of
coexistence densities are quantitatively incorrect.
