I am sorry that I keep harping-on about Penal Substitution. This is not because I think that I am an expert on this topic, rather the opposite - I use this blog simply to process my thoughts, and I have not thought through all the issues that can be raised with this understanding of the atonement.
The Father is not persuaded to show mercy by the Son; rather, the Father sent the Son and they act together. There is no conflict between justice and mercy. The Father is dealing with the mystery of evil and its consequences to deliver sinners. The death of Jesus is not a human sacrifice to enable God to forgive, but the action of God himself who, in his mercy, provides the remedy for sin: it cannot be too strongly emphasized that it is God who suffers on the cross.
... the Father and the Son are acting together in the act of atonement; God bears in himself the dire consequences of sin so that sinners, who are totally unable to save themselves, may be delivered from their sin through faith in the Son of God who loved them and gave himself for them (Rom. 5:8). The doctrine of the Trinity is our firm defense against and false suggestion that God the Father had to be appeased by the Son in order to bring about his purpose of redemption.