In the books it is not explicitly stated that ability to speak with snakes also transcends into control over them.
But in the 2nd year, Harry is able to order a snake to let Justin Finch-Fletchley go, which clearly shows that the ability allows a kind of influence on snakes. Voldemort is also known to be able to control the Basilisk and also to a certain extent apparently Nagini though it's not explicitly mentioned.
The book doesn't suggest that Harry has complete control over the snake. The quote from the encounter with Justin Finch-Fletchley states:
He brandished his wand at the snake and there was a loud bang; the snake, instead of vanishing, flew ten feet into the air and fell back to the floor with a loud smack. Enraged, hissing furiously, it slithered straight toward Justin Finch-Fletchley and raised itself again, fangs exposed, poised to strike.
Harry wasn't sure what made him do it. He wasn't even aware of deciding to do it. All he knew was that his legs were carrying him forward as though he was on casters and that he had shouted stupidly at the snake, "Leave him alone!" And miraculously - inexplicably - the snake slumped to the floor, docile as a thick, black garden hose, its eyes now on Harry. Harry felt the fear drain out of him. He knew the snake wouldn't attack anyone now, though how he knew it, he couldn't have explained.
The snake was ready to attack and Harry shouted at it to stop. I would equate this to using a firm voice with a dog. Even if you are not that dog's owner and it is not familiar with you; generally, if you use a firm voice the dog will obey. It helps too that the snake already "understands" the language that is being spoken.
The relationship with Voldemort and Nagini is slightly different as mentioned by Dumbledore in Half Blood Prince (emphasis mine):
I think he is perhaps as fond of her as he can be of anything. He certainly likes to keep her close and has an unusual amount of control over her, even for a Parselmouth.
It seems clear that even Dumbledore thinks the level of control Voldemort has over Nagini is unusual. Presumably, because Nagini contains a part of his soul, Voldemort has stronger "control" over her. It is also likely that Voldemort has been in the company of Nagini for some time (and possibly has possessed Nagini in the past), which would have strengthened the bond between them.
Finally, the Basilisk was said to be bred to obey only the heir of Slytherin. According to the legend described by Professor Binns in CoS:
“That is believed to be some sort of monster, which the Heir of Slytherin alone can control,” said Professor Binns in his dry, reedy voice.
While I don't have any evidence to support it, my personal guess is that it depends on intentions. For instance, when Harry was at the zoo, the communication was in a conversational manner. However, when he stopped the snake from attacking Justin and when he opened the door to the chamber of secrets, he used more force in his voice. He was panicked with Justin and just snapped it out. I seem to remember him trying and failing to talk to the squiggle to open the door to the chamber of secrets until he said it more forcefully. So I feel that you can control snakes when you put magic force into the command and Harry is untrained, so he only commands them unintentionally.
