There's reason celebrate "Father's" or "Mother's day" but it's been doing recently as you said by modern or western followers celebrate honoring fathers and celebrating parenthood, paternal bonds, and the influence of parents in society.
But I found there is "Sutra of Filial Piety"

The Sutra of Filial Piety (or Sutra on the Profundity of Filial Love, Sutra on Parental Benevolence) is an apocryphal sutra composed in China and apparently an exercise in Buddhist apologetics. It is claimed to have been translated by the monk Kumārajīva.

The text attempts to synthesise native Confucian ideals with Buddhist teachings and was probably produced by Chinese Buddhist monks in imitation of the Confucian Classic of Filial Piety. The sutra seeks to refute Confucian criticism that Buddhism's traditionally monastic focus undermines the virtue of filial piety.

The sutra is still highly popular in China and Japan and in the latter is sometimes used as a focus in Naikan-type introspection practices.

The Filial Piety Sutra

The Buddha Speaks about the Deep Kindness
of Parents and the Difficulty in Repaying it.

Thus I have heard, at one time, the Buddha dwelt at Shravasti, in the Jeta Grove, in the Garden of the Benefactor of Orphans and the Solitary, together with a gathering of great Bhikshus, twelve hundred fifty in all and with all of the Bodhisattvas, thirty-eight thousand in all.

At that time, the World Honoured One led the great assembly on a walk toward the south. Suddenly they came upon a pile of bones beside the road. The World Honoured One turned to face them, placed his five limbs on the ground, and bowed respectfully.

Ananda put his palms together and asked the World Honoured One, "The Tathagata is the GreatTeacher of the Triple Realm and the compassionate father of beings of the four kinds of births. He has the respect and reverence of the entire assembly.
What is the reason that he now bows to a pile of dried bones?

The Buddha told Ananda, "Although all of you are my foremost disciples and have been members of the Sangha for a long time, you still have not achieved far-reaching understanding. This pile of bones could have belonged to my ancestors from former lives. They could have been my parents in many past lives. That is the reason I now bow to them." The Buddha continued speaking to Ananda, "These bones we are looking at can be divided into two groups. One group is composed of the bones of men, which are heavy and white in color. The other group is composed of the bones of women, which are light and black in color."

Ananda said to the Buddha, "World Honoured One, when men are alive in the world, they adorn their bodies with robes, belts, shoes, hats and other fine attire, so that they clearly assume a male appearance. When women are alive, they put on cosmetics, perfumes, powders, and elegant fragrances to adorn their bodies, so that they clearly assume a female appearance. Yet, once man and women die, all that is left are their bones. How does one tell them apart? Please teach us how you are able to distinguish them."

The Buddha answered Ananda, "If when men are in the world, they enter temples, listen to explanations of Sutras and Vinaya texts, make obeisance to the Triple Gem, and recite the Buddha's names, then when they die, their bones will be heavy and white in colour. Most women in the world have little wisdom and are saturated with emotion. They give birth to and raise children, feeling that this is their duty. Each child relies on its mother's milk for life and nourishment, and that milk is a transformation of the mother's blood. Each child can drink up to one thousand two hundred gallons of its mother's milk. Because of this drain on the mother's body whereby the child takes milk for its nourishment, the mother becomes worn and haggard and so her bones turn black in colour and are light in weight."

When Ananda heard these words, he felt a pain in his heart as if he had been stabbled and wept silently. He said to the World Honoured One,
"How can one repay one's mother's kindness and virtue?"

The Buddha told Ananda, "Listen well, and I will explain it for you in detail. The fetus grows in its mother's womb for ten lunar months. What bitterness she goes though while it dwells there! In the first month of pregnancy, the life of the fetus is as precarious as a dewdrop on grass: how likely that it will not last from morning to evening but will evaporate by midday!"

"During the second lunar month, the embryo congeals like curds. In the third month it is like coagulated blood. During the fourth month of pregnancy, the fetus begins to assume a slightly human form. During the fifth month in the womb, the child's five limbs- two legs, two arms, and a head- start to take shape. In the sixth lunar month of pregnancy, the child begins to develop the essences of the six sense faculties: the eyes, ears, nose, tongue, body and mind. During the seventh month, the three hundred sixty bones and joints are formed, and the eighty-four thousand hair pores are also complete. In the eight lunar month of the pregnancy, the intellect and the nine apertures are formed. By the ninth month the fetus has learned to assimilate the different nutrients of the foods it eats. For example, it can assimilate the essence of peaches, pears, certain plant roots and the five kinds of grains."

"Inside the mother's body, the solid internal organs used for storing hang downward, while the hollow internal organs used for processing, spiral upward. These can be likened to three mountains, which arise from the face of the earth. We can call these mountains Mount Sumeru, Karma Mountain, and Blood Mountain. These analogous mountains come together and form a single range in a pattern of upward peaks and downward valleys. So too, the coagulation of the mother's blood from her internal organs forms a single substance, which becomes the child's food."

"During the tenth month of pregnancy, the body of the fetus is completed and ready to be born. If the child is extremely filial, it will emerge with palms joined together in respect and the birth will be peaceful and auspicious. The mother will remain uninjured by the birth and will not suffer pain.
However, if the child is extremely rebellious in nature, to the extent that it is capable of committing the five rebellious acts*, then it will injure its mother's womb, rip apart its mother's heart and liver, or get entangled in its mother's bones. The birth will feel like the slices of a thousand knives or like ten thousand sharp swords stabbing her heart. Those are the agonies involved in the birth of a defiant and rebellious child."

To explain more clearly, there are ten types of kindnesses bestowed by the mother on the child:

The first is the kindness of providing protection and care while the child is in the womb.

The second is the kindness of bearing suffering during the birth. The third is the kindness of forgetting all the pain once the child has been born.

The fourth is the kindness of eating the bitter herself and saving the sweet for the child.

The fifth is the kindness of moving the child to a dry place and lying in the wet herself.

The sixth is the kindness of suckling the child at her breast, nourishing and bringing up the child.

The seventh is the kindness of washing away the unclean.

The eight is the kindness of always thinking of the child when it has traveled far.

The ninth is the kindness of deep care and devotion.

The tenth is the kindness of ultimate pity and sympathy.

1. THE KINDNESS OF PROVIDING PROTECTION AND CARE WHILE THE CHILD IS IN THE WOMB

The causes and conditions from accumulated kalpas grows heavy, Until in this life the child ends up in its Mother's womb. As the months pass, the five vital organs develop; Within seven weeks the six sense organs start to grow. The mother's body becomes as heavy as a mountain; The stillness and movements of the fetus are like a kalpic wind disaster. The mother's fine clothes no longer hang properly, And so her mirror gathers dust.

2. THE KINDNESS OF BEARING SUFFERING DURING BIRTH

The pregnancy lasts for ten lunar months And culminates in difficult labour at the approach of the birth. Meanwhile, each morning the mother is seriously ill And during each day drowsy and sluggish. Her fear and agitation are difficult to describe; Grieving and tears fill her breast. She painfully tells her family That she is only afraid that death will overtake her.

3. THE KINDNESS OF FORGETTING ALL THE PAIN ONCE THE CHILD HAS BEEN BORN

On the day the compassionate mothers bears the child, Her five organs all open wide, Leaving her totally exhausted in body and mind. The blood flows as from a slaughtered lamb; Yet, upon hearing that the child is healthy, She is overcome with redoubling joy, But after the joy, the grief returns, And the agony wrenches her very insides.

4. THE KINDNESS OF EATING THE BITTER HERSELF AND SAVING THE SWEET FOR THE CHILD

The kindness of both parents is profound and deep, Their care and devotion never cease. Never resting, the mother saves the sweet for the child, And without complain she swallows the bitter herself. Her love is weighty and her emotion difficult to bear; Her kindness is deep and so is her compassion. Only wanting the child to get its fill, The compassionate mother doesn't speak of her own hunger.

5. THE KINDNESS OF MOVING THE CHILD TO A DRY PLACE AND LYING IN THE WET HERSELF

The mother is willing to be wet So that the child can be dry. With her two breasts she satisfies its hunger and thirst; Covering it with her sleeve, she protects it from the wind and cold. In kindness, her head rarely rests on the pillow, And yet she does this happily, So long as the child is comfortable, The kind mother seeks no solace for herself.

6. THE KINDNESS OF SUCKLING THE CHILD AT HER BREAST, NOURISHING AND BRINGING UP THE CHILD

The kind mother is like the great earth. The stern father is like the encompassing heaven: One covers from above; the other supports from below. The kindness of parents is such that They know no hatred or anger toward their offspring, And are not displeased, even if the child is born crippled. After the mother carries the child in her womb and gives birth to it, The parents care for and protect it together until the end of their days.

7. THE KINDNESS OF WASHING AWAY THE UNCLEAN

Originally, she had a pretty face and a beautiful body, Her spirit was strong and vibrant. Her eyebrows were like fresh green willows, And her complexion would have put a red rose to shame. But her kindness is so deep she will forgo a beautiful face. Although washing away the filth injures her constitution, The kind mother acts solely for the sake of her sons and daughters, And willingly allows her beauty to fade.

8. THE KINDNESS OF ALWAYS THINKING OF THE CHILD WHEN IT HAS TRAVELED FAR

The death of loved ones is difficult to endure. But separation is also painful. When the child travels afar, The mother worries in her village. From morning until night, her heart is with her child, And a thousand tears fall from her eyes. Like the monkey weeping silently in love for her child, Bit by bit her heart is broken.

9. THE KINDNESS OF DEEP CARE AND DEVOTION

How heavy is parental kindness and emotional concern! Their kindness is deep and difficult to repay. Willingly they undergo suffering on their child's behalf. If the child toils, the parents are uncomfortable. If they hear that he has traveled far, They worry that at night he will have to lie in the cold. Even a moment's pain suffered by their sons and daughters. Will cause the parents sustained distress.

10. THE KINDNESS OF ULTIMATE COMPASSION AND SYMPATHY

The kindness of parents is profound and important. Their tender concern never cease. From the moment they awake each day, their thoughts are with their children. Whether the children are near or far away, the parents think of them often. Even if a mother lives for a hundred years, She will constantly worry about her eighty year old child. Do you wish to know when such kindness and love ends? It doesn't even begin to dissipate until her life is over!

Filial Piety Sutra
by Translated by Upasika Terri Nicholson

There are other suttas speak about "Mother" and "Father"

"Monks, I will teach you the level of a person of no integrity and the level of a person of integrity. Listen & pay close attention. I will speak."

"As you say, lord," the monks responded.........
Kataññu Suttas: Gratitude

And here is answer your question exactly:

This was said by the Blessed One, said by the Arahant, so I have heard: "Living with Brahma are those families where, in the home, mother & father are revered by the children. Living with the first devas are those families where, in the home, mother & father are revered by the children. Living with the first teachers are those families where, in the home, mother & father are revered by the children. Living with those worthy of gifts are those families where, in the home, mother & father are revered by the children.
'Brahma' is a designation for mother & father.
'The first devas' is a designation for mother & father.
'The first teachers' is a designation for mother & father.
'Those worthy of gifts' is a designation for mother & father. Why is that? Mother & father do much for their children. They care for them, nourish them, introduce them to this world."

Mother & father,
compassionate to their family,
are called
Brahma,
first teachers,
those worthy of gifts
from their children.
So the wise should pay them
homage,
honor
with food & drink
clothing & bedding
anointing & bathing
& washing their feet.
Performing these services to their parents,
the wise
are praised right here
and after death
rejoice in heaven.
Itivuttaka: The Group of Fours

I was thinking while wishing my friend a father's day if it is in Buddhism but with your question I got that a lot beyond it.
Thanks a lot.
So even if Modern or western disciples and followers celebrates these fest on an occasion of Days. Buddha said it is responsibility for children to look after their parents.
All he said enchanted.

My father is dead but I try to go the extra mile and help--really help my eldest uncle and those closest to my father in ways that I can whether it is:

• psychologically and emotionally supportive by giving happy thoughts and emotions and not supporting negativity
• virtually supportive (I work in IT so I know most things that are useful) by setting up phone, accounts, etc.
• reminisce about my father's good qualities and actions frequently with family, ignoring the bad