  Parking functions were classically defined for $n$ cars attempting to park on
a one-way street with $n$ parking spots, where cars only drive forward.
Subsequently, parking functions have been generalized in various ways,
including allowing cars the option of driving backward. The set $PF_{n,k}$ of
$k$-Naples parking functions have cars who can drive backward a maximum of $k$
steps before driving forward. A recursive formula for $|PF_{n,k}|$ has been
obtained, though deriving a closed formula for $|PF_{n,k}|$ appears difficult.
In addition, an important subset $B_{n,k}$ of $PF_{n,k}$, called the contained
$k$-Naples parking functions, has been shown, with a non-bijective proof, to
have the same cardinality as that of the set $PF_n$ of classical parking
functions, independent of $k$.
  In this paper, we study $k$-Naples parking functions in the more general
context of $m$ cars and $n$ parking spots, for any $m \leq n$. We use various
parking function involutions to establish a bijection between the contained
$k$-Naples parking functions and the classical parking functions, from which it
can be deduced that the two sets have the same number of ties. Then we extend
this bijection to inject the set of $k$-Naples parking functions into a certain
set of obstructed parking functions, providing an upper bound for the
cardinality of the former set.
