Then said Jesus, Father, forgive them; for they know not what they do. And they parted his raiment, and cast lots. Luke 23:34 KJV
If Jesus has the power to forgive sins, why in this specific case does he resort to the Father?
But that ye may know that the Son of man hath power on earth to forgive sins, (he saith to the sick of the palsy,) Mark 2:10 KJV
And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. Matthew 28:18 KJV
That, "the Son of man hath power on earth to forgive sins" (Mark 2:10) is beyond dispute as the OP has correctly documented. So why did not Jesus simply forgive them? Why was the father involved in such a request from Jesus as recorded in Luke 23:34? Let me suggest several reasons:
Therefore I will allot Him a portion with the great, and He will divide the spoils with the strong, because He has poured out His life unto death, and He was numbered with the transgressors. Yet He bore the sin of many and made intercession for the transgressors.
Others may be able to think of further reasons why Jesus so prayed.
APPENDIX - Sequence of Jesus sayings on the cross
According to the Cambridge commentary, the order of the Jesus "seven last words" on the cross is as follows:
"Father forgive them" ... They were the first of the seven words from the Cross, of which three (Luke 23:34; Luke 23:43; Luke 23:46) are recorded by St Luke only, and three’(John 19:27-28; John 19:30) by St John only. The last cry also began with the word “Father.” The seven words are
Luke 23:34. The Prayer for the Murderers.
Luke 23:43. The Promise to the Penitent.
John 19:26. The provision for the Mother.
Matthew 27:46; Mark 15:34. Eli, Eli, lama sabachthani?
John 19:28. The sole expression of human agony.
John 19:30. “It is finished.”
Luke 23:46. “Father, into Thy hands I commend My spirit.”
He was asking the Father to forgive them for killing Him, knowing that He was going to go ahead and go through with dying.
He was about to not physically be there, so the Father, after His body died, would have to be the one to forgive them, after their sin was fully committed, the sin of killing Jesus.
His last words in Luke show that everything about Him was put into the Father's hands at that moment.
"Father, into your hands I commend my spirit" Luke 23:46
If you were being tortured to death, would you at that moment be able to truly and sincerely forgive your torturers? He wants to, but is unable. That's why he asks his Father.
EDIT - In the light of a comment, I realise that this is a personal hypothesis and not directly backed up by scripture. I will continue to look for evidence.
In the meantime, my revised answer is as follows:
Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven. (Matthew 7:21).
Jesus says here that (regardless of whether he can forgive sins or not) entry into heaven depends on doing the will of the Father.
His tormentors are not following the will of the Father by torturing the Son. He is asking for intercession.
This is the sin against the Son of Mankind. This was the Father's son that was being killed unmercifully, even though He was perfectly innocent. The Son of Mankind was also God the Father's son.
Not only did Jesus forgive them, but it was important to know that God the Father forgave them as well.
Something to think about the suffering the Father bore too. It brings the Fathers pain before us, yet He purposed it for the good of His Son as well as all creation. God was in Christ reconciling the world to Himself.
God was in Christ reconciling the world to Himself, not reckoning their trespasses to them, 2 Corinthians 5:17
He forgave this sin against His Son who represented God Himself. All this anger and judgment of mankind certainly was against God Himself as well. He forgave!
Then said Jesus, Father, forgive them; for they know not what they do. And they parted his raiment, and cast lots. Luke 23:34 KJV
Who were "they"? There should have two groups of people
But Jesus turning unto them said, Daughters of Jerusalem, weep not for me, but weep for yourselves, and for your children. (KJV)
However, for those who participated in Jesus trial before Pontius Pilate, who answered in Matthew 27:25 would not be forgiven.
Matt 27:25 Then answered all the people, and said, His blood be on us, and on our children. (KJV)
Jesus turned to the Father to forgive them, for His power was taken away during His crucifixion, as seen in Mark 15:34
34 And at the ninth hour Jesus cried with a loud voice, saying, Eloi, Eloi, lama sabachthani? which is, being interpreted, My God, my God, why hast thou forsaken me? (KJV)
Simply, because he and the Father are one and work in communion. Your difficulty is from assuming that this did not occur the other times.
Jesus asks the Father to forgive Roman soldiers who are following orders of their superior (also Roman official) [Mt27:27]
Obviously those Romans were nor under the Old Testament neither Mosaik Law and thus could not be charged as sinners [Romans 5:13]. So logically there was nothing to forgive by strictly Mosaik or Jewish authority.
Jesus was not speaking on his own behalf, as though he did not have the power in himself to forgive. Rather, asking the Father to forgive is commensurate with his role as mediator.
His words are those of an advocate (cf 1 Jn 2:1), making the case for God’s mercy based not on the innocence but the ignorance of those who “know not what they do'' (cf Acts 3:17, 1 Cor 3:8, 1 Tim 1:13). His words also constitute a prayer of intercession, in keeping with the exercise of his priestly office (see Gill’s commentary).
But He, because He continues forever, has an unchangeable priesthood. 25 Therefore He is also able to save to the uttermost those who come to God through Him, since He always lives to make intercession for them. – Heb 7:24-25
Jesus’ words are thus in accordance with his role as the one who stands as mediator between God and man. As both advocate and victim, priest and sacrifice, he was in a unique position to do so. It is thus most fitting that the words of Lk 23:34 were addressed to the Father.
Now all things are of God, who has reconciled us to Himself through Jesus Christ, and has given us the ministry of reconciliation, that is, that God was in Christ reconciling the world to Himself, not imputing their trespasses to them – 2 Cor 5:18-19
