Am I required to pay for the sacrificial animal (hady) for my parents who are performing Hajj At-Tamattu‘ at my expense, or is it permissible for them to fast instead of offering the hady due to their insufficient funds, and would this be considered a manipulation on my part?

The son is not obligated to bear the expenses of Hajj for his parents. If he volunteers to do so, he will be rewarded. If his father performs Hajj as Hajj at-Tamattu‘, the son is not obligated to provide the sacrificial animal (hady). If he gives his father the cost of the hady, it is an act of kindness and righteousness. If he does not give it, the father should fast ten days due to his inability to provide the hady. It is advisable to provide the cost of the hady for them, as it completes the act of righteousness and kindness.