Aren't there some striking parallels between the Jesus and Zeus stories?
No. This website makes the following claims:
1. Zeus was born in a cave on Crete. Christ was born in a cave in Israel
It's questionable whether Jesus was born in a cave or not. The Bible doesn't specifically say so, though it is true that caves were sometimes used as mangers.
2. Zeus was the greatest of the gods of Olympus. Christ is the head of the Christian Church
How is this even a comparison? That Zeus was the greatest of gods of Olympus might be a sort of comparison to God the Father, but not to Jesus (God the Son). Whether Jesus is "the head of the Christian Church" is an iffy statement in and of itself, and doesn't compare to what was said about Zeus just before.
3. Zeus was born in a cave to hide him from his father Cronus. Christ was born in a cave to hide him from Herod.
The statement about Jesus is just plain wrong. Jesus and His family were not hiding from Herod on the night Jesus was born. They fled from Herod later on, several months after Jesus' birth.
4. The goat Amalthea and a swarm of bees accompanied the newborn god and offered him nourishment during his early years. According to variations on the myth, other animals which helped Zeus were the eagle, pigeons, the dog and the sow. Animals offered warmth to the baby Jesus in the cave.
There is nothing in the Bible about the animals offering warmth to baby Jesus.
5. The Cretan myths, unlike those of the rest of Greece, relate that Zeus died and was reborn each year. Zeus Cretagenes, the Cretan-born Zeus, is thought to have replaced an older deity, the Cretan god of vegetation, who died in Autumn and was reborn in Spring, following the eternal cycle of nature. Christ was crucified, died and was resurrected.
I'm finding several websites talking about Zeus' death in Cretan myths, but none saying that he died and was reborn each year. If anyone can find me an example of such a myth, I would love to see it. Even if true, a death and rebirth is a reincarnation, not a resurrection, so it's not much of a comparison to Jesus, especially since the statement involves a death in autumn and rebirth in spring, while Jesus' death and resurrection were only a few days apart.