Can an elderly parent pay rent to an adult child if he/she lives in the child's home?

The agreement should be notarized, and written in very simple, straightforward verbiage. I speak from recent experience. I entered into a written agreement with my elderly mother (pre-dementia). In the past several months, she has begun believing people are conspiring against her. The list of conspirators includes her doctors, her neighbors, and sadly, me. She has claimed that I "tricked" her into signing our agreement. Fortunately, it was a simple, three paragraph agreement written in terms so straightforward a fourth-grader could comprehend its meaning. Also, it was notarized by a notary. There is no way to predict how a person's brain will change over time, and it's best to protect yourself by anticipating what is least expected.

Is it then considered 'income?'

Simple answer -- yes. Draw up a contract of how much your father will pay toward rent and upkeep each month, so there will be a paper trail of the amount of money being paid to you. The amount, of course, should be reasonable.