Police said they were called to the property in Regent Crescent, Skipton at about 7.20am on Tuesday.
Officers found the bodies of a man in his 60s and a woman in her 70s in the house.
A man in his 30s, who was known to the victims, was arrested at the scene on suspicion of murder. He remains in custody, North Yorkshire Police said.
Det Insp Steve Menzies, of the Major Investigation Team, said: “Local residents will understandably be shocked to hear about this tragic incident.
“Although the investigation is at an early stage, a man has been arrested, and we are not currently looking for anyone else in connection with the deaths.”
Police family liaison officers were providing support to those affected, he added.
He said officers were conducting reassurance patrols in the area and urged anyone with information who could assist the investigation to get in touch.
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