The most popular myth holds that Dionysus was the son of Zeus by
Semele - who would become the goddess Thyone.
Zeus was the youngest son of Kronos (Time) and Rhea (Flow), who
was the the youngest Titan son of Ouranos (Sky Dome) and Gaea
(Earth); Ouranos was the son of Gaea, and Gaea either emerged from
creation second, or was the offspring of Aither (Bright Upper Air)
and Hemera (Day) who were born of Erebus (Darkness) and Nyx (Night)
the children of Chaos (Air) who was the first being to emerge from
creation.
Now for Semele, her father was Kadmos, and her mother the
goddess Harmonia - the daughter of Aphrodite and Ares.
Ares was the son of Hera (and Zeus), the sister-wife of Zeus, so
his line follows that of Zeus.
Aphrodite is either the daughter of Zeus and Dione - a Titan
sister of Kronos, or she was born of the sea foam caused by the
blood of Ouranos at his castration.
Kadmos *(Cadmus), was a Phoenician prince, the son of King
Agenor and queen Telephassa of Tyre.
Telephassa is either the daughter of Nilus the god of the Nile
and Nephele - either a cloud nymph that Zeus created; or a daughter
of Oceanus, the Titan son of Gaea and Ouranus. Nilus also being the
son of the Titans Oceanus and Tethys.
Agenor is the son of either Poseidon (brother of Zeus) and
Libya, the daughter of Epaphus the King of Egypt, a son of Zeus and
Io - Io being the daughter of the river god Inakhos, a son of the
Titans Oceanus and Tethys. Melia/Argia, Io's mother, was a daughter
of Oceanus.
Agenor's other set of parents are Belus (said to be his brother
in the other myth of his parentage) and Achiroë/Anchirrhoë/Anchinoë
his mother, a naiad daughter of Nilus, the river god of the
Nile.