The Father did not experience the cross like Jesus, but He did suffer in His new experience of separation from Christ. This is an initial introductory discussion on this truth.
Let's set the scene
We read how all things were created by God and that Jesus as the Logos was co-creator with the Father:
In the beginning was the Word [Logos], and the Word was with God, and the Word was God. He was in the beginning with God. All things were made through Him, and without Him nothing was made that was made.
Then, we read of the Logos becoming man.
And the Word became flesh and dwelt among us, and we beheld His glory, the glory as of the only begotten of the Father, full of grace and truth.
Jesus is the Logos in the flesh. He is not a created being. But, instead, as Paul put it He is:
God was manifested in the flesh.
1 Timothy 3:16 (See Note below)
This existence of Jesus before the world was made, he mentions in his words of prayer, where He tells of both, the Father loving Him before the foundation of the world and, also their having and sharing glory together.
And now, O Father, glorify Me together with Yourself, with the glory which I had with You before the world was . . . for You loved Me before the foundation of the world.
John 17:5, 24
These, of course, are great pointers to the joint deity of Jesus with the Father, especially the shared glory, since God made it clear in the Hebrew Scripture that:
I am the LORD, that is My name; And My glory I will not give to another, Nor My praise to graven images.
If our minds could begin to gather, to appreciate a little, the notion of these two Persons of the godhead (these words are limited in scope, but let's use them here to gain a little of the picture of this existence): these TWO being together, forever in perfect unity and harmony, before Creation and then together involved in Creation where we read in the Genesis account:
Then God said, "Let Us make man in Our image, according to Our likeness. . ."
The pronouns "Us . . . Our . . . Our" all serve to point to the plural Persons involved in creation.
Forever these TWO have never been apart, they are ONE, as Jesus said,
I and My Father are one.
However, on the cross, during his crucifixion Jesus cried out:
My God, My God, why have You forsaken Me?
For a period of time, while on the cross, the Father and the Son were separated.
They both experienced something they never knew before. The pain of being apart was felt by both. This division, this chasm, was suffered by the Father and by the Son.
Jesus was alone and felt alone, but the Father also felt the separation from the Son. The Father personally felt this and no other can appreciate the full meaning of this, since no one has been united with another person to the same extent of time or intensity, as the Father and the Son have from time before the foundation of the world.
What we do know is they both faced the reality of this moment of eventual separation and planned for it from the beginning. They both knew they would suffer and experience this loss. But, they still went ahead with Creation, in the certain knowledge this was worth it for a future joy:
Jesus . . . for the joy that was set before Him endured the cross, despising the shame...
As Jesus said,
. . . so must the Son of Man be lifted up, that whoever believes in Him should not perish but have eternal life.
Both the Father and the Son knew that in order to have this Family of Light in eternity, it was necessary to pay for it through this eventual separation, so all who believed could then be together with them: united forever. The "Father of lights" (James 1:17) and "the light of the world" (John 8:12) separated for a time to enable the possibility that the Family of "lights" (Philippians 2:15) be established.
A possible meaning
Now, since this suffering, this separation is so foundational, the cross is central to the existence of the universe both in terms of time and place: Indeed, were there no earth and heavens (including all space and stars) and just this separation event, even with the angels and spiritual hosts (of all kinds) present, then it was a planned central tenet of redemption for all beings God created that would fall, but wanted to return. A fall, which is a departure from the light, cannot be restored to original communion with the source of all light, unless the mathematical and poetic balance of equation of the consequence of the departure is fully put in place to enable such a return: this is what the cross does. Or, rather what happened while Jesus was on the cross does. The cross was just the place where it happened.
In seeing this dilemma, this mathematical problem if you wish, God planned in creation for this event to happen and was prepared for this separation which they would both fully endure. This is why we read of this event:
Him [Jesus], being delivered by the determined purpose and foreknowledge of God, you have taken by lawless hands, have crucified, and put to death;
God created man with skin on the outside and red liquid just beneath its surface, to demonstrate his gift of life to all. But He did not leave things there:
whom God raised up, having loosed the pains of death, because it was not possible that He should be held by it.
"Who..." or "He..." as words in the translation of 1 Timothy 3:16 are found in manuscripts which other versions are based on; "God..." is in the Codex Alexandrinus and Codex Ephraemi and quoted by Ignatius (1st century), Barnabas (130s), Hyppolytus (170-235), Dydimus (313-398), Gregory of Nyssa (335-395), and Chrysostom (347-407) - this core information is gleaned from page 137 of The King James Version Defended by Edward F. Hills (an advocate for the Received Text) published by Eye Opener Publishers ISBN 0-915923-00-9.