Epiphenomenalism suffers from quite a few logical flaws. 

For one, it contradicts Newton's law of reaction, in that it implies that there could exist cases where causation is asymmetric and non-reversable: X can cause Y but Y cannot cause X. To my knowledge we have yet to find any evidence of such asymmetric causation in nature. What seems to be the case is that if X can impact Y, then Y can impact X. Causality works both ways. 

More fundamentally, something that exists *as part of this world* must exist in some sense as a cause for something else. It must make a difference in this world, otherwise in what sense is it part of it? How could we even ascertain its existence? How could we notice it? 

Epiphenomena do not exist, as defined. They have no way of existing, other than as a figment of our imagination.