Is it permissible for my father to say to my mother: "Half of the house is hers if he dies before her"?

Your father's statement: "If I die before you, half the house is yours," is what is known as 'ruqba' (life tenancy), and there is disagreement among jurists regarding it.

Imam Shafi'i, Imam Ahmad, and Abu Yusuf believe that ruqba is permissible, and the condition is nullified because ownership is the basis. This is also the madhhab of the Hanbalis, who hold that the ruqba belongs to the one for whom it was made, and the condition is nullified. They cite as evidence the hadith of Zayd ibn Thabit and Ibn Abbas, may Allah be pleased with them both, that the Prophet, peace and blessings be upon him, said: "Do not grant your properties as ruqba, for whoever grants something as ruqba, it belongs to the one for whom it was granted."

As for Abu Hanifa and Muhammad, ruqba is invalid, and it is a loan ( 'ariyya ) which the grantor may revoke. The Malikis stated that ruqba is not permissible.

In summary: The condition is nullified, and the gift to the one for whom it was made is valid.