How can I escape the suffering of losing my father?

Your father is not his body, he is information, "spirit"(~attitude, character) - right? As you interact with him, this information enters your system. When your father dies, it will continue in you. Not grieving over your father, not clinging to his solid form, you will allow his qualities to freely manifest in yourself. So in a way you will become continuation of your father, of what was the best in him. In that sense, your father would never die. If you cling to his solid form, you will only block the process of "him" continuing in "you".

If your father was a good father to you (doing good karma), according to the Pali suttas, he will have a good rebirth. The Pali suttas say:

These beings — who were endowed with good conduct of body, speech & mind, who did not revile noble ones, who held right views and undertook actions under the influence of right views — with the break-up of the body, after death, have re-appeared in the good destinations, in the heavenly world.

MN 4

You need to eventually face this suffering due to your attachment. Such attachment is usually deep seated.

But what may be helpful is to contemplate on anicca or impermanence.

For this, you can take A Walk in the Woods with Phra Khantipalo.

Also, read the story of Patacara in Dhammapada 113.

This is a difficult problem. Although an arahant (enlightened person) would be free of grief in this situation, for the rest of us, sorrow is more or less inevitable. There are a number of approaches. One is to practice mindfulness of feelings, realizing that your feelings change from moment to moment. Another is to realize that your father possesses the three marks of existence: Suffering, Impermanence and Egolessness.

1. Like all other beings, his life is full of suffering, and to grasp onto him is to cause suffering.
2. His life is impermanent, which you have already shown some awareness of. Being mindful now of this unavoidable fact will help you deal with it when the time comes.
3. He is not a self and does not possess a self. He is composed of many things which themselves are composed of many things, and these things are always changing.

As mentioned by others, a regular meditation practice is also helpful for developing serenity and insight.

From the Dhammapada:

Sabbe sankhara anicca ti. yada pannaya passati; atha nibbindati dukkhe. esa maggo visuddhiya

Every thing (experience specifically, since that is the primary) is ephemeral. And by experientially internalising that wisdom, one comes out of suffering. This is the path of purification.

Practice the Dharma. Whether our father goes first or we go first, ultimately we have to face impermanence one day. If our father goes first, we feel grief naturally, that is because we are attached to something-our father. Yet if we go first, we are presented with a huge fear, because of the unknown of death. Buddhism deals with all these grief and fear- sufferings. Thus one should practice Buddhism diligently and without doubt because we really don't want all these sufferings.

Try to behave virtuously, do what you can, and see the virtue of/in others.

Even when you want the world to be other than it is, I think it would be worse if on top of that you regretted your own past misbehaviour; conversely you may find it better if you had good relations.

I think that "recollection of virtue" (including generosity) appears frequently (e.g. here), in one form or another, as something of a consolation or support.

There was a story of a very accomplished monk in the Soto Zen tradition. One day, he heard that his master had died. He took the news with apparent ease. At his funeral, he was composed - greeting well wishers, accepting their condolences, smiling, and even laughing. This lasted for much of the memorial service. Right before they were to take the body away for cremation, this monk when to the side of his masters coffin. He began to sob uncontrollably. All of the people in attendance began to look at him with mouths agape. Realizing that he was being stared at and wise to the reason why - after all it's not often that you see a Zen master lose his composure so fully and unashamedly - the monk said to the onlookers:

"My master, who I have known all of my life, who I loved as deeply as my own father, has died. If I want to cry, I'm going to cry."

Buddhist practice isn't Vulcanism. It isn't about transcending our emotion and sitting in a tepid path of imperturbability. Rather, Buddhist practice is about being openhearted. It is about becoming fully and unashamedly intimate with the world. If your father dies, of course you will suffer. Why shouldn't you? Allow yourself to experience the full extent of your grief. Mourn wholeheartedly. Anything short of that is a disservice both to your relationship with him and to your own feelings of love.

Let go of what is going to cause the suffering. I.e. your attachment towards him. Practice Satipattana meditation.

The way you express your feeling reminds me of a sutta where Buddha told story of how gods in heaven tremble in fear when they heard Buddha preached about impermanance. Or even a great horse sees a shadow of a whip. Since this a Q&A, Anatta is a cure.