It refers to a dick. The kid sits astride the stick, with the wheels on the ground behind him and the head held up in front. The stick is then in just the right position to give the horse it's name. I'd never heard of them being called stick horses 'till I went looking for one on the web.

We learned it in school when we were kids (it seems like 460 years, but it's not quite that long ago;-). I guess by the time we learned the meaning, we were too old to play with one anyway (cock horse, that is).

If it offends your sensibilities feel free to sing..

'Ride a stick horse to Banbury cross' ;-)