Is it permissible for a father to build on a part of the public road that he had annexed to his house and paid a fine for?

If the fine paid by your father was the price of the plot of land from the competent authority, then construction is permissible. However, if it was merely a fine for a violation and encroachment, then the land remains public property and construction upon it is not permissible, because public rights are more sacred than private ones. The Sharia has forbidden the unlawful taking of people's property, as stated in His (Allah's) Almighty saying: "O you who have believed, do not consume one another's property unlawfully but [only] that your properties may be exchanged by mutual approval." And whoever wrongs [others by taking] a handspan of land, Allah will encircle him with it from seven earths on the Day of Resurrection.