What is the ruling of Islamic law on holding a mixed-rite wedding ceremony, which includes the removal of Hindu religious manifestations from it, to please non-Muslim parents, knowing that both parties are Muslim?

You have done well by refusing the Hindu wedding ceremony, as it involves the presence and endorsement of falsehood. An Islamic wedding ceremony does not have a specific form, but it must be free from objectionable practices. Among these objectionable practices are the intermingling of men and women, and imitating Hindus in their adornment or special clothing, such as the wife sitting on a chair surrounded by fruits and food, with non-mahram (definition) men present to feed her, and wearing "Bollywood" (Bindi), which is a special adornment for Hindus, on the head. If the men are not your mahrams, then this is forbidden intermingling. As for adornment, if it is specific to Hindus, then it is not permissible to wear it, because the Prophet, peace and blessings be upon him, said: "Whoever imitates a people is one of them." You should strive to convince your parents to hold a ceremony free from objectionable practices.