I think it's certainly possible. I even had a theory on him you can find here:
https://harrypotter.fandom.com/f/p/3258895044706361229
Here's a copy of the main text (with some minor updates 😄):
As we see with Harrys example, the ability to speak Parseltongue can be bestowed once a being has been made a Horcrux (even by accident) All that seems to be required is that the original wizard remains alive. It would have been interesting if Quirrell was able to speak Parseltongue after being possessed by Voldemort- as described in the wiki: 'effectively turned into a living Horcrux for the last year of his life'.
If that were true.. do you think the ability to speak Parseltongue could be sign that the wizard has been possessed by Dark wizard who speaks Parseltongue, who continues exists as a roaming spirit or has new body due to having at least one Horcrux.
Following this example- could Herpo_the_Foul have possessed various wizards through time as he searched for a way to return to a body? Did he perhaps possess Salazar Slytherin? Is this why Salazar was able to create a Basilisk? With Herpo still wandering around, did Salazar retain the ability to speak parseltongue through fatherhood? If Herpo remains alive, would this be passed to Salazars children? Is Slytherins heirs ability to speak Parseltongue a clue that Herpo is still 'alive'?