On Parmenides and Epicurus, Reason and Belief

Something that is vitally important, is to remember whose tradition the Epicureans belonged to. They were not heirs to the Platonist tradition, no. No rather, Epicurus was from a succession of thought and debate that extended back through Pyrrho, through Democritus, deeply influenced by the rich thought of Empedocles, ultimately reaching back into the questions of Parmenides.

And who was Parmenides? He wrote a grand poem, which knocked the Pythagoreans off their own rock of belief, the belief in pure mathematics that lead to their mysticism.

Why did the Epicureans believe, why were they anti-sceptics and sceptics at the same time?