Alan runs into his father's office before realizing that he mistook a squatter for his father. The squatter immediately notes that it's cold out and offers him hot coffee and warm clothing.
Judy is horribly close to death at the end of the game, having been poisoned. She says that she misses her dead parents (something she has been denying throughout the movie)—probably because she realizes she's going to see them in a few moments.
Judy: I... wish mom and dad were here...
Alan tries to emulate his father, then realizes that he's being a jerk, and apologizes to Peter.
Nora's behavior towards the end of the movie. Although she seemed to have a hard time taking care of Judy and Peter, she became concerned about them when she realized that something must have happened to them. Even after she and Carl get swept away by the monsoon, she made her back into the destroyed house and was calling out for her niece and nephew.
Young Alan looking at the pieces that belonged to Judy and Peter, who aren't even born yet.
The ending where after meeting Judy, Peter and their parents, Alan and Sarah take measures to protect said parents despite the fact that neither of the kids remember them at all.
At the end, his father, despite having no way of knowing that Alan went through 26 years of hell, apologizes to Alan for his own outburst and is willing to talk to him, man to man. Alan hugs his father, who cracks a brief smile before hugging his son back and making amends.
Alan: I'm so glad you're back! Sam: I've only been gone a few minutes. Alan: It feels like a lot longer to me. [...] Sam: We'll talk later: Man to man! Alan: How about father and son?
Even better is when Young Alan reveals to his father that he was the one who put the shoe in the machine and got Carl fired, Mr. Parish doesn't even get mad, in fact he seems proud that Alan owned up to his mistake. He even smiles as he looks at Sarah sitting in the living room.
"Oh, my God, there they are..."
After the quicksand floor is solid again:
Sarah: I was afraid I'd lost you again. Alan: Thanks for sticking around this time.
The sheer joy on Alan's face when he thanks Peter for getting him out of Jumanji.
Carl singing with the other guests at the Christmas party; thanks to Alan confessing he accidentally destroyed Carl's shoe, Carl didn't get fired and drift away from the Parrish family. A deleted scene revealed Carl was the president of the company, no doubt thanks to his shoe design.
What is Alan doing during the party? He is on the phone with his dad, happily talking about a new shoe he is putting out and telling him he can't wait to see him tomorrow at the airport and how it will be the best Christmas ever. It's clear Alan and his father rebuilt their relationship and are now very close.
The fact that Alan was willing to help Judy and Peter when the siblings get sucked into Jumanji.
Even though Judy is angry at Alan for putting Peter in danger, she still agrees to go to Jumanji to help Alan. What makes it better is how happy she and Peter were seeing that Alan was all right.
In the episode "No Dice", Peter mentions that Alan wouldn't have to worry about him and Judy, and Alan expresses concern for what happens to them in the real world, such as whether or not the children are doing well in their new school. This indicates that Alan cares about the children outside of Jumanji.
When Judy gives Alan a family heirloom as a birthday present, saying that it belongs in the family. Alan's reaction to the gift is what really makes it heartwarming. Before that, Judy was driven to tears as after she begs Alan to survive by remembering his family, the images show that his family are Judy and Peter.
The Grand Finale where after Alan finally escapes Jumanji, he and Nora meet each other again and to the kids and the audiences' surprise, they instantly fell in love. Judy and Peter can't help but smile because it means Alan might become their uncle. Judy and Peter might not have their parents, like in the film, but after spending forty episodes with him, Alan became a father to them. So they don't mind.