The Fubini numbers (aka ordered Bell numbers) F_n (A000670) count the number of ways n runners can place in a race, with ties. In this talk, we consider two refinements of F_n: first, by grouping placements by the number of pairs of runners where the runner with the higher bib finishes before a runner with a lower bib; second, by how many runners tie at each place. The former refinement leads to a polynomial refinement of F_n and the latter leads to a symmetric function refinement of F_n. We derive simple generating functions and recurrence relations for both refinements. Moreover, we show that the symmetric function results implies the polynomial results through a general theory on word enumerators. We further view these results in the same sphere of results of Konheim–Weiss (1966) and Haiman (1994) on parking functions by interpretting Fubini words as unit interval parking functions of Harris–Hadaway (2021) and Bradt et al (2024).