The Ellis County Sheriff's Office says the siblings shared a house with their father, who has Alzheimer's disease.
Justice of the Peace Jackie Miller Jr. said Wednesday that the father was at the home near Venus, about 30 miles southwest of Dallas, when the killings happened Sunday morning.
Miller says 49-year-old Diane Stallings was strangled and beaten. Authorities believe 47-year-old Doil McAnally Jr. then fatally shot himself.
Miller says the father apparently was unaware of the deaths. The woman's boyfriend discovered the bodies several hours later.
Authorities are trying to determine why the siblings apparently argued.
Other relatives are caring for the father.