Yes, the Bible does command children to honor, obey and respect their parents. (Eph. 6:1, 2) But nowhere does it advocate the commemoration of a special "Mother’s Day".
Jehovah's Witnesses avoid participating in any celebrations with non-Christian religious origins. Some may say Mother's Day does not have roots in ancient paganism and that it is presently considered a largely secular event. But the earliest Mother's Day celebrations can be traced back to the spring celebrations of ancient Greece in honor of Rhea, the mother of the gods. On the origin of such observance, the Encyclopædia Britannica states:
“A festival derived from the custom of mother worship in ancient Greece. Formal mother worship, with ceremonies to Cybele, or Rhea, the Great Mother of the Gods, were performed on the Ides of March throughout Asia Minor.”—(1959), Vol. 15, p. 849.
Regarding the adoption of Mother’s Day in the United States, the New York Times of May 10, 1953, reported:
“In spite of the popularity of Cybele, . . . and sporadic occasions honoring mothers during the Middle Ages, it was not until 1914 that the proper combination of sentimentality, idealistic promotion and hard business sense impelled the United States Congress to designate the second Sunday in May as Mother’s Day.”
And according to the Website "Mothers Day Central" (which appears to have an unbiased view of Mother's Day), it says:
"Only recently dubbed “Mother's Day,” the highly traditional practice of honoring of Motherhood is rooted in antiquity, and past rites typically had strong symbolic and spiritual overtones; societies tended to celebrate Goddesses and symbols rather than actual Mothers. In fact, the personal, human touch to Mother’s Day is a relatively new phenomenon. The maternal objects of adoration ranged from mythological female deities to the Christian Church itself. Only in the past few centuries did celebrations of Motherhood develop a decidedly human focus."
Further adding to this, the website Mother's Day 123 Holiday, under the heading "Mother's Day History", agrees by saying:
"Contrary to popular belief, Mother's Day was not conceived and fine-tuned in the boardroom of Hallmark. The earliest tributes to mothers date back to the annual spring festival the Greeks dedicated to Rhea, the mother of many deities, and to the offerings ancient Romans made to their Great Mother of Gods, Cybele. Christians [?] celebrated this festival on the fourth Sunday in Lent in honor of Mary, mother of Christ. In England this holiday was expanded to include all mothers and was called Mothering Sunday."
So in view of the false religious background of Mother’s Day, is it not clear that Christians in the first century would not have commemorated these days? So, then, is it not right to shun such observances today and thus obey the Bible’s command to “quit touching the unclean thing”? (2 Cor. 6:17) If a holiday or custom is being deliberately participated in by a Christian, it must have absolutely no known pagan religion associations.
For more, see:
What is the origin of the practice of setting aside a day to honor mothers? (rs p. 176-p. 182; Watchtower Online Library)
Paganism - Links to Information (Search For Bible Truths)
Holidays - Links to Information (Search For Bible Truths)