Let $b^{p}_{\alpha}$ , $0<\alpha \le1$ , be the parabolic Bergman space, the Banach space of solutions of parabolic equations $(\partial/\partial t+(-\varDelta)^{\alpha})u=0$ on the upper half space $\mathbf{R}^{n+1}_{+}$ which have finite
$L^{p}$ norms. We study Carleson type measures
on $b^{p}_{\alpha}$ , and give a necessary and
sufficient condition for a measure $\mu$ on $\mathbf{R}^{n+1}_{+}$ to be of Carleson type
on $b^{p}_{\alpha}$ . As an application, we
characterize bounded Toeplitz operators in the
space $b^{2}_{\alpha}$ .