The [Dark Mark] symbol is a skull, with a serpent coming out of the mouth, like a tongue. This obviously symbolises Parselmouths and their ability to talk to snakes, in other words, them being "serpent-tongued." Voldemort was the man who gave rise to the Death Eaters and formed the symbol . . . he chose that as the symbol because he considered himself the heir of Slytherin. Yet, the meaning actually goes back even further than that.
Inside the Chamber of Secrets, the basilisk's lair is Salazar Slytherin's statue. When Tom Riddle releases the basilisk from its lair, [the snake comes of out the statue's mouth]. A snake coming out of the mouth of a skull. The Dark Mark. Coincidence? Hell no.
— Quora user Sathya Iyer