Can God the father forgive sins without Jesus playing any part?
Jesus Christ is the incarnation of the eternal word/wisdom of God (
Logos means both word and wisdom), and we would say that God is never without His word/His wisdom. So the question does not make sense as asked. "Can God forgive without His wisdom playing any part?" is not a question that makes sense to ask about God, because that assumes that God forgives without having any divine sense of what He Himself is doing or why. That is not the case, even if we humans do not always understand His ways or His thoughts.
The closest possible analogue I can think of in Islam, while admittedly imperfect, is the status of the Qur'an, which as I understand it is considered by Muslims to
preexist its revelation to Muhammad. So since it is with God already, if someone was to ask you if Muhammad was
really necessary for it to exist, you'd probably consider the question malformed, as it does not take into account the preexistent nature of the speech (the Qur'an being God's speech, correct?). And yet, if Muhammad (or say, any human) is not involved, so it never gets revealed in the form you have now known as the Qur'an, then how do you know anything of it in the first place? Sure, you could say "Well, humans are naturally inclined to submission to God, so we would still have the pre-Islamic prophets and hanifs to show us the right way to whatever degree", but those things are not the Qur'an. The Qur'an is a specific revelation (or rather, a series of them) given to Muhammad, not to other people at other times. There are reasons why Islam exists as its own religion separate from Judaism, Christianity, Sabianism, etc., and it is not respectful to the history of the belief to treat things as though they are somehow interchangeable just because they
could have gone differently in some other world than the one that actually exists, where Muhammad did claim to receive such revelations, and those revelations were eventually collected into the text of the Qur'an.
So could God the Father forgive without
the incarnate Jesus having ever walked the earth (which I suspect is how you meant that question)? I should hope so, as He forgave the sins of the Ninevites in the time of Jonah after accepting their repentance and fasting, to use but one example from the Old Testament. It's not like that didn't happen just because Jesus was not incarnate at that time. But remember what I wrote at the beginning of this post: this does
not mean that the wisdom of the God was somehow not involved.
What Al Masihi has been posting about the cooperation of the Persons of the Holy Trinity in the redemption of man (including obviously the forgiveness of sins) is correct. The Persons of the Holy Trinity do not act in isolation from one another, but cooperatively. This has to do with their internal relation to one another, which is eternally present because God is eternal (the Son and the Holy Spirit were not 'added later', but have always been God).
God's revelation to man is a somewhat different topic, since it is inherently occurring within time, as humanity invented time and exists within time and space.