This is something I have always pointed out to fellow fans, only to be shot down. I believe that Harry certainly had the power to control the basilisk, just as Ginny was able to while she was under Tom Riddle's influence. I think it simply comes down to the fact that this idea never occurs to him. It is important for readers to remember that Harry's ability to speak parseltongue was a newly discovered talent. If he had been more seasoned, perhaps he might have thought to try.
Slytherin's heir only had an advantage over others when it came to snakes because of the communicative power between them. The snake still maintains autonomy and free will; it simply chooses to follow the wizard's bidding more often than not, because the parseltongue can understand and command him. Proof of this can be found in book 6 when Morphin, who is also a parselmouth, is singing to a snake. He says something along of the lines of "you be good to Morphin or he'll nail you to the door". There is in fact a snake already nailed to his door, suggesting that a previous snake behaved aggressively towards him despite Morphin's ability to communicate with it.
That in mind, Salazar Slytherin was taking an awful risk in assuming that just because the monster he left behind was a snake, that it would follow his heir's bidding. It could have just as easily finished the heir off and went on a rampage, killing pure bloods and muggles alike. Now that, would have been a plot twist.