From Connolly and Dodge, The Ancient City
This is a drawing of part of the Parthenon frieze depicting an apobatês ('dismounter') a fully armed warrior who rode in a chariot for a while, dismounted, and ran alongside the chariot for a time, and then jumped back onto the chariot. The event is not depicted on any surviving Panathenaic vase, so it is probable that there was a different prize for this contest, but it is not known what the prize was. The apobatês race is depicted on the Parthenon frieze although, as far as we know, it was not part of the procession. It is probably present on the frieze to represent all the athletic contests that were an important part of the festival. This race was held in high regard as this passage from an essay attributed to Demosthenes (Pseudo-Demosthenes, Eroticus, 61.23-29):
Scholars have speculated that this contest was a commemoration of Homeric warfare, in which warriors used chariots to ride to and return from the battlefield. Perhaps, the importance of the apobatês race evident in the quote above and in its prominence on the Parthenon Frieze (11 chariots with charioteer and apobatês) was because the invention of the chariot was attributed to Erichthonios, the ancestor of all Athenians.…you have singled out the noblest and grandest of competitive exerrcises and the one most in harmony with your natural gifts, one which approximates the realities of warfare through the habituation to martial weapons and the laborious effort of running in the magnificence and majesty of the equipment simulates the might of the gods, presents besides the most delectable spectacle, embraces the largest number and the greatest variety of features and has been deemed worthy of the most valuable prizes. (N.W. De Witt translation)