Detectives believe the boy's father, Tony Easthope, 34, shot his former girlfriend, Lucy Carter, 23, with his own rifle before turning the gun on himself. Their bodies were found in Mr Easthope's grey Metro car by a neighbour of Ms Carter's behind the row of houses where she lived in Scunthorpe early on Saturday morning.
The gun was near the car and a post-mortem examination revealed that both had died from gunshot wounds.
Police said the couple had lived together for several years but the relationship had been "stormy" since they split up six months ago and Ms Carter moved to a house in a nearby street with their son, Ethan.
On Friday night, she left Ethan with her sister to go out to a nightclub with friends.
Ms Carter then caught a taxi to a friend's house and is believed to have left there in another taxi at around midnight to return home.
A police spokeswoman said: "It is believed Lucy died first - there is no suggestion she took her own life.
"We believe Tony died after Lucy and was killed by a single shot from his own gun. Numerous forensic tests are needed to confirm this but at present we are not looking for anyone else in connection with these tragic deaths."
Mr Easthope, a steelworker, was a registered firearms holder and a member of a gun club. The couple's son is being cared for by Ms Carter's mother.
Superintendent Quentin Dowse, leading the police investigation, said: "On Friday night Lucy went out with some friends and met Tony by chance in a bar in Scunthorpe.
"Later they met again by chance at a nightclub in the town. She left with a girlfriend around 11 and then went to the girlfriend's house and got a taxi home about midnight.
"There is then a gap until 8am on Saturday when they were found in his car parked at the back of her house."
One of Ms Carter's neighbours said: "I knew Lucy to say hello to and she was always a very pleasant and nice girl. I have heard though that he asked her to marry him on New Year's Eve and she had said `no'."