What is the history behind the names " Chinese Parsley" and "Cilantro"?
The term Chinese parsley appears to have originated in the United States near New York City in the late 19th century. The OED's first citation is from the New York Times in 1895, although Google books finds an 1893 reference, discussing the same Chinese-American farms mentioned below.
This 1899 Google books reference explains how Hen Shang started a farm in Astoria (in Queens, NYC) around 1880, where he grew Chinese vegetables. After he proved successful, he was joined in partnership by some other Chinese immigrants, and several others started their own farms. One of the crops mentioned was Chinese parsley. Other crops were yun tsoi, bok tsoi, and Chinese eggplant.
So it seems probable that many New Yorker's first exposure to cilantro came from the cilantro grown on these Chinese farms in Astoria. It looks something like parsley; hence the name.
The names coriander and cilantro are both ultimately derived from the Latin word coriandrum, which in turn comes from ancient Greek name for the herb, κορίαννον (koríannon), coming through French and Spanish, respectively.