If your parent has alzheimers, please check on their cat during each visit. If you don’t live nearby, ask the caretaker to check on the cat.
This may seem like too much to ask. You may be exhausted, both emotionally and physically. Still, I ask you to do this for the sake of the pet who your parent loves.
Here is a true story about what can happen.
A woman with Alzheimers wanted to clean her cat, so she used Chlorox bleach on the cat’s privates. Her intentions were good. She meant no harm.
A parent with Alzheimers (or any other serious illness) should not be expected to make logical and safe decisions about cat care. It is simply not realistic.
This is a tough situation. All one can do is act in the way that the parent would have wanted you to act, if s/he had full cognitive function.