(יב) כַּבֵּ֥ד אֶת־אָבִ֖יךָ וְאֶת־אִמֶּ֑ךָ לְמַ֙עַן֙ יַאֲרִכ֣וּן יָמֶ֔יךָ עַ֚ל הָאֲדָמָ֔ה אֲשֶׁר־ה' אֱלֹהֶ֖יךָ נֹתֵ֥ן לָֽךְ׃ (ס)
(12) Honour thy father and thy mother, that thy days may be long upon the land which the LORD thy God giveth thee.
This pasuk is stating that every day should be Father's Day. 365 days a year, 7 days a week, 24 hours a day. We still need this day to remind us that we respect, love and fear our parents.
(טז) כַּבֵּ֤ד אֶת־אָבִ֙יךָ֙ וְאֶת־אִמֶּ֔ךָ כַּאֲשֶׁ֥ר צִוְּךָ֖ ה' אֱלֹהֶ֑יךָ לְמַ֣עַן ׀ יַאֲרִיכֻ֣ן יָמֶ֗יךָ וּלְמַ֙עַן֙ יִ֣יטַב לָ֔ךְ עַ֚ל הָֽאֲדָמָ֔ה אֲשֶׁר־ה' אֱלֹהֶ֖יךָ נֹתֵ֥ן לָֽךְ׃ (ס)
(16) Honour thy father and thy mother, as the LORD thy God commanded thee; that thy days may be long, and that it may go well with thee, upon the land which the LORD thy God giveth thee.
As it is stated again, to remind us again to respect, love and fear our parents. This commandment of the Ten Commandments is in the first 5, the section of ben adam la'makom [between man and Hashem]
The Ten Commandments are read during the winter and summer while Father's and Mother's Day are in the spring [with Father's Day on the border]. We have almost the whole year reminding us of the rule of Kibbud Ab ve'Em.
I highly suggest readnig the article ''ABCs of Honoring Parents" on aish.com