I too doubt very much the Tsarevich would have been allowed to ride a horse at parades. But he did participate at them with his father. For instance, the year before, in 1911, both reviewed the Moskovsky Guards Infantry Regiment, on foot, for its jubilee. The regiment had the Tsarevich as its colonel-in-chief. Both father and son can be seen reviewing the regiment in a newsreel which can be seen on the National Geographic tape “Russia’s last Tsar”. Unfortunately, the proud colonel-in-chief wasn’t allowed to drink the health of his regiment as his father can be seen doing so while Alexis is standing next to him, no doubt wishing he would be allowed to have a drink too!
As for the Tsarevich’s first independent engagement at Borodino, I haven’t found anything about it either. Nicholas II’s diary and detailed letters to his mother describing the events might provide the answer. Unfortunately I only have a few excerpts. Princess Eugénie’s book doesn’t mention anything special except a funny anecdote about Alexis when the IF stopped at Smolensk on their way back: he managed to approach the buffet at the House of the Nobility and drink some champagne without anyone noticing. He became quite talkative, made jokes, laugh aloud and had a marvelous time with the ladies! His parents only found out the reason why he had suddenly lost his shyness when they were back on the train: still giggling, Alexis told them of his conversations with the ladies of Smolensk and said he could hear his stomach rumble while talking to them. I guess the “colonel-in-chief” had finally got his revenge for the year before!