I am curious to know if during their numerous encounters with one another, could Harry Potter and Lord Voldemort have talked to each other using Parseltongue if they had chosen to do so? Or does Parseltongue only work between humans and snakes?
Parseltongue definitely does work as a method of conversing between humans. This can be seen in Chapter Ten of Half-Blood Prince when Morfin Gaunt speaks Parseltongue to Bob Ogden:
“You understand him, I’m sure, Harry?” said Dumbledore quietly.
“Yes, of course,” said Harry, slightly nonplussed. “Why can’t Ogden — ?”
But as his eyes found the dead snake on the door again, he suddenly understood.
“He’s speaking Parseltongue?”
“Very good,” said Dumbledore, nodding and smiling.
In this instance Ogden couldn’t understand it, not being a Parselmouth, but shortly thereafter Marvolo Gaunt speaks Parseltongue to Morfin, which Morfin does understand:
Mr Gaunt spoke out of the corner of his mouth to Morfin.
“Get in the house. Don’t argue.”
This time, ready for it, Harry recognized Parseltongue; even while he could understand what was being said, he distinguished the weird hissing noise that was all Ogden could hear.
In fact, later in Chapter Seventeen, Voldemort himself speaks Parseltongue to Morfin:
“YOU!” he bellowed. “YOU!”
And he hurtled drunkenly at Riddle, wand and knife held aloft.
“Stop.” Riddle spoke in Parseltongue.
So we definitely know that it can be done, and that Voldemort could do it. As for whether Harry could do it that is less clear. In theory he should be able to do it, but Harry notoriously had difficulty speaking Parseltounge when not talking to a snake. For instance, in Chapter Sixteen of Chamber of Secrets:
“Harry,” said Ron. “Say something. Something in Parseltongue.”
“But —” Harry thought hard. The only times he’d ever managed to speak Parseltongue were when he’d been faced with a real snake. He stared hard at the tiny engraving, trying to imagine it was real.
“Open up,” he said. He looked at Ron, who shook his head.
“English,” he said.
Harry looked back at the snake, willing himself to believe it was alive. If he moved his head, the candlelight made it look as though it were moving.
“Open up,” he said.
Except that the words weren’t what he heard; a strange hissing had escaped him, and at once the tap glowed with a brilliant white light and began to spin.
However, in Chapter Nineteen of Deathly Hallows Harry was able to use Parseltongue to open the locket with no issue at all, so perhaps he had progressed in his Parseltongue abilities:
“One . . . two . . . three . . . open.”
The last word came as a hiss and a snarl and the golden doors of the locket swung wide with a little click.
If that is the case, he might have also been able to use it to speak to another person, though considering that he wasn’t acquainted with any other Parselmouths besides Voldemort he wouldn’t have had much opportunity to do so.
From the Harry Potter Wiki Parseltongue can be spoken and understood by all Parselmouths. However, the two speakers need to be highly experienced with speaking the language.
Aside from serpent-based creatures, Parselmouths can communicate with each other with the language, as Harry understood Tom Riddle's commands to the Serpent of Slytherin, and the House of Gaunt communicate with each other almost exclusively in Parseltongue. While inherited, Parseltongue usually requires the speaker to face a snake-based creature or object shaped like a snake (e.g. a carving); more proficient speakers may be able to speak it at will, as the Gaunt family was very proficient in the language — enough so for Morfin to speak to Bob Ogden with it, while Voldemort could speak to the Statue of Slytherin to release the basilisk.
So Harry and Voldemort can speak to each other in Parseltongue with little to no help.