In my opinion, there ARE parallels with Hellenistic mystery religions but usually in terms of narrative topoi and far less in specific detail, where OT sources are much more obvious (especially in the narrative gospels). It is true that some popular writers are less careful in presenting parallels and often "improve" them by making the ANE parallels closer to the story/stories of Jesus than they really were (see the link Narkissos provided for examples of this circularity). The oft repeated claim that Horus was born in December 25 with shepherds, who was baptized by An the Baptizer, and who was crucified, is a classic example of this. So one should be careful to go back to primary sources as much as possible, and critically examine the merits of each purported parallel which vary in terms of representativeness. Our knowledge of many ANE mystery religions are also fragmentary (such as that of Mithra, which varied between East and West) and one must be careful to avoid circularity again if the gaps are filled in by other sources. Finally, there is not a single Christ myth or story, but many different traditions. The class "Christ story" that is usually used for comparison picks and chooses a compromise single story from these.
BTW, I think the most convincing mythic parallel is one not usually mentioned by popularizing "Christ myth" writers, and that is the Baal myth. That was a native, indigenous Israelite mythic tradition, and I believe it played a more central role in the development, inasmuch as it is found in muted form in the OT itself.