We would like to know based on Qur'an and Hadith, is it allowed for husband and wifeto see the private parts during intercourse?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

It is permissible for both the husband and wife to look at each other's private parts for sexual desire or not; Imam Tirmizi narrated that a person said: "I asked the Prophet; from whom should we conceal our private parts and to whom can we expose them? He replied, 'Conceal your private parts except from your wife or your slave woman". Moreover a husband could seek pleasure from his wife's vagina, thereupon he can see it also just as it is permissible for him to see her whole body. Since it is permissible for him to touch and have sex with her, then looking at her vagina is also permissible for him.

But it is desirable not to see her vagina since the Prophet did not do it. As Aisha said: "The Prophet died; he never looked at me (private parts) and I never looked at his". If this Hadith is sound it teaches only noble character, but does not establish the prohibition for the above acts.

Allah knows best.