The English term parallel comes from a Greek term meaning "beside each other." It is the placing of two things side by side to compare them. Comparison is one of the basic tools that scholarship uses, whether placing two manuscripts side by side or two texts or two cultures. Usually doing so allows one to see similarities and differences. When the differences vastly outnumber the similarities scholars might subjectively determine that the parallel is stretched, or far-fetched, or non-existent. What the parallels mean can be discussed, or even debated. For example, both the ancient Egyptians and the ancient Mayans depicted victorious kings trampling enemies under their feet. The parallels are clearly there, but what they mean is something that is perhaps worth discussing. (Personally, I have not found additional evidence that suggests that this particular parallel could be anything more than an interesting coincidence, but I might be missing something.)
When it comes to scripture, however, Latter-day Saints are encouraged to draw parallels:
for I did liken all scriptures unto us, that it might be for our profit and learning. (1 Nephi 19:23)Nephi encourages his readers to liken what they read (or what he reads to them) to themselves as something that is profitable and has a potential to learn from. One could argue that Nephi encourages parallelomania as a scriptural hermeneutic. Thus there is a disconnect between those who would forbid Latter-day Saints the drawing of parallels between sources and situations and what the scriptures themselves tell Latter-day Saints they should do. Cries of parallelomania in LDS scripture study are inappropriate and bear an extra burden of proof on those making the accusation.