The author's parents are correct. There is nothing "Christian" about New Years. I have no idea where that idea came from. At best, you could make the claim that it's a Gentile holiday and thus celebrating is would be a kind of assimilation, but it's not a religious holiday in any way, shape, or form. The Christian "New Year" begins on the fourth Sunday before Christmas. January 1 *IS* a holy day on the Christian calendar, but it's the Feast of the Circumcision (also called the Feast of the Holy Name) - because it's the 8th day after Christmas Day ;) New Year's celebrations are about the transition to a new secular year. Folks don't party until midnight to commemorate Jesus's bris.
from the article: Confessions of a Russian-New-Year's-celebrating Orthodox Jew