What do Father's and Mother's Days Got to do With Hinduism?

No true Hindu celebrates the "Father's Day" or "Mother's Day". If one, however does, obviously, one is not a Hindu. 

If someone insists that he is a Hindu, and yet honours Father's or Mother's Day, and celebrates the day with his father or mother, he is either a thoroughly confused Hindu or plainly a hypocrite who passes himself off as a member of the Sanatana Dharma. 

Even in the lands where these days originated, i.e., in the West, no serious practitioner of the mainstream religions equates the days with any religious or even spiritual significance; they consider them vulgar and inane. Compared to them, innumerable misguided "Hindus" on the Indian subcontinent, in Singapore, Malaysia and many Western countries have not only been celebrating the said days but have also been conducting "special prayers" in Hindu temples for goddesses and gods on Mother's Day and Father's Day respectively. That is the acme of confounded foolishness and ignoble ignorance.

To a true Hindu, one who practises the tenets of the ancient religion, every living day is a father's day, and every breathing moment is a day to celebrate his mother's sacrifice. In some scientific quarters, it has been asserted that there are 50 trillion cells in the human body, yet others have claimed that it is between 50 and 100 trillion --- whatever the numbers be, every one of the cells that we have in us owes its existence to our parents, whether they are dead or alive. 

They strive and survive because of our parents who brought us to this world. Therefore, to a Hindu, every moment, every second, every nanosecond of our existence is a tribute to our parents, is a celebration in honour of their memories, their sacrifice, love and compassion. Every living moment of ours is sacred, and is dedicated to our parents who were instrumental in bringing us to this world to fulfil our life's journey. 

To celebrate their penance for but one day is, therefore, a serious denigration of their sacrifice. There is no Father's or Mother's day for a Hindu. Every day --- every living day --- is a Hindu's parents' day. He, therefore, does not designate a day of the year to remember his parents, to respect and pay obeisance to them. No one sets aside a day to breathe; similarly, to a practising Hindu, Father's or Mother's day has no meaning. 

There is an arrant horde of hypocrites who blithely ignore their parents for an entire year, but make  perfunctory telephone calls to them on Father's or Mother's Day to wish them, or visit them in the Homes where the old people languish in the memory of the children that they brought up all their lives. Celebration of Father's or Mother's day is such children's effort to assuage their guilty conscience that pricks them away for their unfilial karma. How deplorable is it to see people who call themselves "Hindus" ape such a tradition that has no place in their religion.

Sources: 

https://www.sanatan.org/en/a/102038.html

http://vedicilluminations.com/downloads/Apology/Mother's%20Day%20Hindu%20Style.txt