Wiktionary
n. 1 The invocation of a person's race in a discussion in order to score points in that discussion. 2 A listing of all horses to be run in a horse race.
Wikipedia
Playing the race card is an idiomatic phrase that refers to exploitation of either racist or anti-racist attitudes by accusing others of racism.
Usage examples of "race card".
I picked up a spare race card that was lying on the table and knew at once why Charlie had paid me such acute attention: two of the very top hurdlers were engaged in battle and the papers had been talking about it for days.
I bought a race card and found a space on the parade-ring rails, and watched the horses for the first race being led around.
I bought a race card and found a space on the parade ring rails, and watched the horses for the first race being led round.
Okay, so just for the record, this was Carol Moseley Braun urging someone not to play a race card.