What is the Sharʿī ruling on my mother abandoning prayer, knowing that she is elderly, suffers from urinary incontinence, and is semi-paralyzed?

It is not permissible for your mother to abandon prayer, even if her condition is as you described, because the Maliki school of thought holds that the exit of سلس ( سلس in this context refers to a continuous discharge, like incontinence) from the private parts does not invalidate wudu’ if it is constant for most or all of the time. Therefore, you can perform wudu’ for your mother once, and her wudu’ will not be invalidated by what exits from her due to illness. If you are not present, she can perform tayammum (definition) by having some dust brought to her to perform tayammum with.