The question: If Jesus has authority on earth to forgive sins, why does He ask the Father to do it in Luke 23:34?

In the New Testament world, the relationship between God, Jesus, and the believer is often illuminated by the cultural concept of a reciprocal, or client-patron, relationship. God the Father is the patron; the believer is the client; Jesus is the Mediator (analogous to a cultural broker) -- who serves as the bridge between the two, acting upon the divine authority He shares with the Father. (In the Old Testament, God would be the suzerain, Moses the mediator, and Israel the vassals; this mirrors the New Testament relationship as well.)

In both instances, Jesus operates as the unique Mediator of God's forgiveness, revealing two aspects of His salvific work: Luke 23:34 is the Mediator's intercession to the Father on humanity's behalf -- an example of Jesus exercising His high priestly role. Luke 5:24 represents the Mediator exercising His divine authority on earth to declare forgiveness directly to the client.

First-century Jews understood such reciprocal relationships perfectly. YHWH (suzerain/patron), Moses (mediator of the covenant), Israel (clients).

Most importantly, the personnel of worship are clearly defined: the Father bestows benefaction on His clients by means of Jesus, the Mediator. Jesus, in turn, intercedes for the concerns of the clients to the Father.