After a long day of trying to beat Green Arrow and his kids in various competitions, as soon as the supervillains attack, Batman orders Alfred to protect the Arrow kids alongside his own, and charges into battle shoulder to shoulder with G.A.
After Tim had lost the kid's competition, Batman was fully intending to lecture him. But when Tim came running over, cheerfully holding his third place medal and saying how he got to third place with the help of his dad, Batman instead patted him on the head, telling him "I'm proud of you, Tim."
Also, while everyone else was busy with the games, baby Terry wanders off and bumps into some villains. Fortunately, Mama Bear Catwoman comes to the rescue and it's Terry's happy squeal of "Mama!" when he recognizes her immediately that does it.
From Father's Day, Alfred hurries a late night working Batman into bed so his kids could "surprise" him. Then, Alfred finds out that Batman wasn't the only father to get something.
After a long day with the kids and their usual bickering, Batman still finds time to tuck them into bed.
In "Holidays with the Justice League", Batman is shown cross-armed and scowling in all of the pictures... except for Father's Day.