Dad should first consider the age of the kid. For very young kids, he should avoid really terrifying details. He can also give the kid a sense of security during the story, like saying 'but this can't happen in our house' or 'I will always protect you from such things'.
He should give the story a happy ending. This way, the kid will not be left with a feeling of dread. For example, if the story is about a haunted house, at the end, the family in the story can make the house a friendly place by being kind to the ghosts.
Once upon a time, Dad and Ada went into an old, abandoned house. The moment they stepped in, the door slammed shut behind them. They heard strange whispers all around. Dad held Ada's hand tightly. As they walked further, they saw a shadowy figure at the end of the hallway. Ada screamed, but Dad tried to be brave. They quickly turned around to find another way out, only to discover that the rooms seemed to be changing and twisting around them. It was a nightmare they couldn't escape.
Sure. There was a kid who moved into an old house. Every night, he heard strange scratching noises coming from the attic. One night, he couldn't take it anymore and went to investigate. When he opened the attic door, a cold wind blew out. He saw a pair of glowing eyes in the darkness. Terrified, he ran back to his room and hid under the covers. He never went back to the attic again.
There was a little girl who walked in the forest near her home. She got lost and as the sun started to set, the forest became very spooky. She heard whispers all around her. Then, she saw a shadowy figure that seemed to be following her. She ran and ran until she found her way home, shaken by the experience. Since then, she never went into the forest alone.
A kid might scream at a scary story because they are not used to dealing with fear in that way. Their little hearts can't handle the sudden rush of adrenaline that comes from hearing something really scary. It could also be that they have no experience with such intense emotions, and the only way they know how to respond to the fear the story creates is by screaming.