  We recall that unit interval parking functions of length $n$ are a subset of
parking functions in which every car parks in its preference or in the spot
after its preference, and Fubini rankings of length $n$ are rankings of $n$
competitors allowing for ties. We present an independent proof of a result of
Hadaway, which establishes that unit interval parking functions and Fubini
rankings are in bijection. We also prove that the cardinality of these sets are
given by Fubini numbers. In addition, we give a complete characterization of
unit interval parking functions by determining when a rearrangement of a unit
interval parking function is again a unit interval parking function. This
yields an identity for the Fubini numbers as a sum of multinomials over
compositions. Moreover, we introduce a generalization of Fubini rankings, which
we call the $r$-Fubini rankings of length $n+r$. We show that this set is in
bijection with unit interval parking functions of length $n+r$ where the first
$r$ cars have distinct preferences. We conclude by establishing that these sets
are enumerated by the $r$-Fubini numbers.
