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

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.

Is it then considered 'income?'

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.