Should he do Hajj on behalf of his father who does not have any money with which to perform Hajj?

Praise be to Allah.

If the situation is as described, that your father is healthy and cannot do Hajj because he is financially unable, then he is not obliged to do Hajj, because Allah says (interpretation of the meaning): “And Hajj (pilgrimage to Makkah) to the House (Ka‘bah) is a duty that mankind owes to Allah, those who can afford the expenses (for one’s conveyance, provision and residence)” [Aal ‘Imraan 3:97]. So it is not valid for you or anyone else to do Hajj on his behalf. But if you can afford to pay for his Hajj, it is prescribed for you to help him so that he can do Hajj for himself. 

And Allah is the Source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his family and companions. End quote. 

Standing Committee for Academic Research and Issuing Fatwas 

Shaykh ‘Abd al-‘Azeez ibn ‘Abd-Allah ibn Baaz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allah ibn Ghadyaan, Shaykh ‘Abd-Allah ibn Qa’ood.