The Greek word, patēr, appears 413 times in the New Testament.
In Luke 15, the Parable of the Prodigal Son, patēr is the word for the father who sees his son a long way off and runs to him.
Open Mounce's Complete Expository Dictionary of Old and New Testament Words, available with Bible Gateway Plus, to that word and here's what you find:
patēr refers to a male parent (e.g., Mk 9:24) or one’s ancestors (e.g., Mt 3:9; Jn 4:12; Acts 7:4, 11, 12, 14) or as a term of respect for an older man (Acts 7:2). It is also used to refer to God, who is the Father of all believers (Mt 5:16) and of Jesus (Mt 7:21; Mk 11:10; Lk 1:32).
Later in the same entry, Mounce traces patēr directly to the Parable of the Prodigal Son:
Jesus compares human fathers to God as the perfect Father (Lk 11:11–13), and God’s identity as Father stands behind the famous parable of the prodigal son in Lk 15.
That's the key. The father in the parable isn't just a character. He is a portrait and a reflection of God—running, embracing, restoring His children.
Mounce traces patēr even further across the Gospels:
In creating a new “family” of believers, Jesus recognizes that his followers will be mothers, brothers, and sisters to each other, but he reserves the role of “father” for God alone (cf. Mt 23:9).
The father who runs in Luke 15 is a portrait of the only One Jesus reserved that title for.
