The Gemmorah in Kiddushin (30b) brings the idea of the analogy between honoring Hashem and honoring one's father (and mother).
"ת"ר נאמר (שמות כ, יב) כבד את אביך ואת אמך ונאמר (משלי ג, ט) כבד את ה' מהונך השוה הכתוב כבוד אב ואם לכבוד המקום
The Sages taught that it is stated: “Honor your father and your mother” (Exodus 20:11), and it is stated: “Honor the Lord with your wealth” (Proverbs 3:9). In this manner, the verse equates the honor of one’s father and mother to the honor of the Omnipresent, as the term “honor” is used in both cases.
וכן בדין ששלשתן שותפין בו ת"ר שלשה שותפין הן באדם הקב"ה ואביו ואמו בזמן שאדם מכבד את אביו ואת אמו אמר הקב"ה מעלה אני עליהם כאילו דרתי ביניהם וכבדוני
And so too, the equating of one’s attitude toward his parents to his attitude toward God is a logical derivation, as the three of them are partners in his creation. As the Sages taught: There are three partners in the forming of a person: The Holy One, Blessed be He, who provides the soul, and his father and his mother. When a person honors his father and mother, the Holy One, Blessed be He, says: I ascribe credit to them as if I dwelt between them and they honor Me as well.
And many many more analogies. The Gemmorah in Brochos 58a teaches Birkat Hashvakh on seeing a king "בא"ה אמ"ה שחלק מכבודו ליראיו".
Why this (or a similar) Brocho is not used for seeing/honoring the father, whose honor also "stems" from Hashem's?
Interestingly, some say that the blessing on a king stems from his ability to sentence to death (see blessing-on-a-king, but both father or Rabbi "can freely" kill a kid (הלכות רוצחים, they are not allowed but exempt from punishment)
Note, this is different from saying Birkat Hamitzvot on the Mitzvah of honoring, which is not said "because" the father can forgive his respect.