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≥n parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter ℓ≥0, we then consider the subset of interval rational parking functions in which each car parks at most ℓ spots away from their initial preference. We call these ℓ-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers m≥n and ℓ. We also establish formulas for the number of nondecreasing ℓ-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between ℓ-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing ℓ=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 ℓ=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.