Green’s Dictionary of Slang

red dog on a white horse n.

also red knight on a white horse

(US black) a woman having her menstrual period.

[US]E. Folb Runnin’ Down Some Lines 157: There are a variety of expressions [...] that refer to a women’s menstrual period – [...] to ride the white horse, the red dog/red knight is on the white horse.
[US](con. 1940s) Deuce Ofay Productions ‘The Jive Bible’ at JiveOn.com 🌐 Don’t you fuck wit’ momma when she be chillin’ wit’ de red knight on de white horse, boy!