  In this paper we study the asymptotic behavior of a random uniform parking
function $\pi_n$ of size $n$. We show that the first $k_n$ places
$\pi_n(1),\dots,\pi_n(k_n)$ of $\pi_n$ are asymptotically i.i.d. and uniform on
$\{1,2,\dots,n\}$, for the total variation distance when $k_n = o(\sqrt{n})$,
and for the Kolmogorov distance when $k_n=o(n)$, improving results of Diaconis
& Hicks. Moreover we give bounds for the rate of convergence, as well as limit
theorems for some statistics like the sum or the maximum of the first $k_n$
parking places. The main tool is a reformulation using conditioned random
walks.
