  Interval parking functions are a generalization of parking functions in which
cars have an interval preference for their parking. We generalize this
definition to parking functions with $n$ cars and $m\geq n$ parking spots,
which we call interval rational parking functions and provide a formula for
their enumeration. By specifying an integer parameter $\ell\geq 0$, we then
consider the subset of interval rational parking functions in which each car
parks at most $\ell$ spots away from their initial preference. We call these
$\ell$-interval rational parking functions and provide recursive formulas to
enumerate this set for all positive integers $m\geq n$ and $\ell$. We also
establish formulas for the number of nondecreasing $\ell$-interval rational
parking functions via the outcome map on rational parking functions. We also
consider the intersection between $\ell$-interval parking functions and Fubini
rankings and show the enumeration of these sets is given by generalized
Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the
set of $1$-interval rational parking functions with $n$ cars and $m$ spots are
in bijection with the set of barred preferential arrangements of $[n]$ with
$m-n$ bars. This readily implies enumerative formulas. Further, in the case
where $\ell=1$, we recover the results of Hadaway and Harris that unit interval
parking functions are in bijection with the set of Fubini rankings, which are
enumerated by the Fubini numbers.
