Is it permissible for my father to perform tayammum for prayer, given that he is ill and unable to purify himself from urine or rise from bed?

The Sharia (definition) has made capacity the criterion for obligation, as Allah Almighty says: "Allah does not burden a soul except with what He has given it." And among the jurisprudential rules is that hardship brings ease, and when a matter becomes difficult, it expands. If your father is able to perform ablution without harm, then he must do so, even with assistance. If he is unable, he should perform tayammum (definition). He is not obligated to remove impurities due to the difficulty of it, and he should pray in the condition that he is able to.