What is the main conflict in the story?

I am guessing you want to know about the conflict of whether Phaeton believed his father was, in fact, Apollo. In order to prove this, he asked Apollo - who was his father - allows him to drive the chariot which pulls the sun through the sky. When Phaeton cannot control the horses, he puts the Earth in danger of being burned. Zeus cannot allow this to happen and has to kill Phaeton with a thunderbolt. The conflict is Phaeton against the other gods along with the hubris (overweening pride) Phaeton feels once he knows that Apollo is his father.