Praise be to Allah.
It is not permissible for the father to see his son’s fiancée, because she is a non-mahram to him. It is only permissible for his son to see her with special permission from sharee‘ah, in order to create harmony and affection after marriage. But no exception has been narrated with regard to the man’s father; in that case the original prohibition on seeing her remains in effect, and she is like any other woman who is a non-mahram to him. So he is enjoined to avert his gaze from her, not to be alone with her and not to shake hands with her.
But when his son does the marriage contract with his fiancée, she becomes his wife, even if he has not consummated the marriage with her, and even if in your country she is still called a fiancée after doing the marriage contract. This customary naming of the matter does not affect anything, so long as the son’s marriage contract has been done. In that case it becomes permissible for the father to look at his son’s wife, because she has become a mahram to him, as Allah, may He be exalted, says (interpretation of the meaning): “Forbidden to you (for marriage) are: your mothers, your daughters... the wives of your sons who (spring) from your own loins” [an-Nisa’ 4:23].
And Allah knows best.