  In the parking model on $\mathbb{Z}^d$, each vertex is initially occupied by
a car (with probability $p$) or by a vacant parking spot (with probability
$1-p$). Cars perform independent random walks and when they enter a vacant
spot, they park there, thereby rendering the spot occupied. Cars visiting
occupied spots simply keep driving (continuing their random walk). It is known
that $p=1/2$ is a critical value in the sense that the origin is a.s. visited
by finitely many distinct cars when $p<1/2$, and by infinitely many distinct
cars when $p\geq 1/2$. Furthermore, any given car a.s. eventually parks for $p
\leq 1/2$ and with positive probability does not park for $p > 1/2$. We study
the subcritical phase and prove that the tail of the parking time $\tau$ of the
car initially at the origin obeys the bounds \[ \exp\left( - C_1
t^{\frac{d}{d+2}}\right) \leq \mathbb{P}_p(\tau > t) \leq \exp\left( - c_2
t^{\frac{d}{d+2}}\right) \] for $p>0$ sufficiently small. For $d=1$, we prove
these inequalities for all $p \in [0,1/2)$. This result presents an asymmetry
with the supercritical phase ($p>1/2$), where methods of Bramson--Lebowitz
imply that for $d=1$ the corresponding tail of the parking time of the parking
spot of the origin decays like $e^{-c\sqrt{t}}$. Our exponent $d/(d+2)$ also
differs from those previously obtained in the case of moving obstacles.
