The $\alpha$-parabolic Bergman space
$\bm{b}^p_\alpha$ is
the set of all $p$-th integrable solutions $u$ of the equation
$(\partial/\partial t + (-\Delta)^{\alpha})u = 0$
on the upper half space, where $0 < \alpha \leq 1$
and $1 \leq p \leq \infty$.
The Huygens property for the above $u$ will be obtained.
After verifying that the space
$\bm{b}^p_\alpha$ forms a Banach space,
we discuss the fundamental
properties. For example, as for the duality,
$(\bm{b}^p_\alpha)^* \cong \bm{b}^q_\alpha$ for $p > 1$ and
$(\bm{b}^1_\alpha)^* \cong \mathcal{B}_\alpha/ \mathbf{R}$
are shown, where
$q$ is the exponent conjugate to $p$ and $\mathcal{B}_\alpha$ is
the $\alpha$-parabolic Bloch space.