In the middle ages of the last century, and in the middle of this country, gymkhana was thought of as a smaller, tighter course that might involve a 3-point turn into or out of a "parking spot" and other novel challenges, all meant to test piloting skills. Autocross was the term for longer-legged, faster courses, which is where 'parking lot racing' seemed to go over time. At least some of the change must have been due to the fact that the relatively little and lightweight european cars were good at gymkhana but were supplanted over time by bigger, faster cars. Classic case in point would be the Tiger which originated as the nimble but mild 4-cylinder Sunbeam Alpine. [We never saw an AC Ace until it showed up as a ground-pounding Cobra.] Minis could do it all, thus proving themselves, literally, exceptional.
"Gymkhana" used to be done on horseback: an exhibition of equestrian skill. I imagine it was not done very much here in the States. We do still have dressage and western trick riding, which might be traceable descendants of gymkhana. And I guess the rodeo events of barrel-racing and calf-roping might squeak in as analogous to autocross.
There were time/speed/distance rallyes in those days too, even out there in the boonies. I gather a few survive these days near our left and right shores. And in fact, some folk in Minnesota are starting one up this summer, God love 'em. Sounded like it might be a fine time; I ran across the posting here in Events.