  We recall that the $k$-Naples parking functions of length $n$ (a
generalization of parking functions) are defined by requiring that a car which
finds its preferred spot occupied must first back up a spot at a time (up to
$k$ spots) before proceeding forward down the street. Note that the parking
functions are the specialization of $k$ to $0$. For a fixed $0\leq k\leq n-1$,
we define a function $\varphi_k$ which maps a $k$-Naples parking function to
the permutation denoting the order in which its cars park. By enumerating the
sizes of the fibers of the map $\varphi_k$ we give a new formula for the number
of $k$-Naples parking functions as a sum over the permutations of length $n$.
  We remark that our formula for enumerating $k$-Naples parking functions is
not recursive, in contrast to the previously known formula of Christensen et al
[CHJ+20]. It can be expressed as the product of the lengths of particular
subsequences of permutations, and its specialization to $k=0$ gives a new way
to describe the number of parking functions of length $n$. We give a formula
for the sizes of the fibers of the map $\varphi_0$, and we provide a recurrence
relation for its corresponding logarithmic generating function. Furthermore, we
relate the $q$-analog of our formula to a new statistic that we denote
$\texttt{area}_k$ and call the $k$-Naples area statistic, the specialization of
which to $k=0$ gives the $\texttt{area}$ statistic on parking functions.
