  We introduce a generalization of parking functions called $t$-metered
$(m,n)$-parking functions, in which one of $m$ cars parks among $n$ spots per
hour then leaves after $t$ hours. We characterize and enumerate these sequences
for $t=1$, $t=m-2$, and $t=n-1$, and provide data for other cases. We
characterize the $1$-metered parking functions by decomposing them into
sections based on which cars are unlucky, and enumerate them using a Lucas
sequence recursion. Additionally, we establish a new combinatorial
interpretation of the numerator of the continued fraction $n-1/(n-1/\cdots)$
($n$ times) as the number of $1$-metered $(n,n)$-parking functions. We
introduce the $(m,n)$-parking function shuffle in order to count
$(m-2)$-metered $(m,n)$-parking functions, which also yields an expression for
the number of $(m,n)$-parking functions with any given first entry. As a
special case, we find that the number of $(m-2)$-metered $(m, m-1)$-parking
functions is equal to the sum of the first entries of classical parking
function of length $m-1$. We enumerate the $(n-1)$-metered $(m,n)$-parking
functions in terms of the number of classical parking functions of length $n$
with certain parking outcomes, which we show are periodic sequences with period
$n$. We conclude with an array of open problems.
