  Recall that $\alpha=(a_1,a_2,\ldots,a_n)\in[n]^n$ is a parking function if
its nondecreasing rearrangement $\beta=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq
i$ for all $1\leq i\leq n$. In this article, we study parking functions based
on their ascents (indices at which $a_i<a_{i+1}$), descents (indices at which
$a_i>a_{i+1}$), and ties (indices at which $a_i=a_{i+1}$). By utilizing
multiset Eulerian polynomials, we give a generating function for the number of
parking functions of length $n$ with $i$ descents. We present a recursive
formula for the number of parking functions of length $n$ with descents at a
specified subset of $[n-1]$. We establish that the number of parking functions
of length $n$ with descents at $I\subset[n-1]$ and descents at $J=\{n-i:i\in
I\}$ are equinumerous. As a special case, we show that the number of parking
functions of length $n$ with descents at the first $k$ indices is given by
$f(n, n-k-1)=\frac{1}{n}\binom{n}{k}\binom{2n-k}{n-k-1}$. We prove this by
bijecting to the set of standard Young tableaux of shape $((n-k)^2,1^k)$, which
are enumerated by $f(n,n-k-1)$. We also study peaks of parking functions, which
are indices at which $a_{i-1}<a_i>a_{i+1}$. We show that the set of parking
functions with no peaks and no ties is enumerated by the Catalan numbers. We
conclude our study by characterizing when a parking function is uniquely
determined by their statistic encoding; a word indicating what indices in the
parking function are ascents, descents, and ties. We provide open problems
throughout.
