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Parking functions were classically defined for $n$ cars attempting to park ona 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 beenobtained, 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, tohave the same cardinality as that of the set $PF_n$ of classical parkingfunctions, independent of $k$.  In this paper, we study $k$-Naples parking functions in the more generalcontext of $m$ cars and $n$ parking spots, for any $m \leq n$. We use variousparking function involutions to establish a bijection between the contained$k$-Naples parking functions and the classical parking functions, from which itcan be deduced that the two sets have the same number of ties. Then we extendthis bijection to inject the set of $k$-Naples parking functions into a certainset of obstructed parking functions, providing an upper bound for thecardinality of the former set.

Connecting $k$-Naples parking functions and obstructed parking functions via involutions