  For any integers $1\leq k\leq n$, we introduce a new family of parking
functions called $k$-vacillating parking functions of length $n$. The parking
rule for $k$-vacillating parking functions allows a car with preference $p$ to
park in the first available spot in encounters among the parking spots numbered
$p$, $p-k$, and $p+k$ (in that order and if those spots exists). In this way,
$k$-vacillating parking functions are a modification of Naples parking
functions, which allow for backwards movement of a car, and of $\ell$-interval
parking functions, which allow a car to park in its preference or up to $\ell$
spots in front of its preference. Among our results, we establish a
combinatorial interpretation for the numerator of the $n$th convergent of the
continued fraction of $\sqrt{2}$, as the number of non-decreasing
$1$-vacillating parking functions of length~$n$. Our main result gives a
product formula for the enumeration of $k$-vacillating parking functions of
length $n$ based on the number of $1$-vacillating parking functions of smaller
length. We also give closed formulas for the number of $k$-vacillating parking
functions of length $n$ which are monotonically increasing or decreasing. We
conclude with some directions for further research.
