In "The Swimming Pool"; Fred ends up getting thrown in jail on the night of his birthday due to a botched misunderstanding (specifically, mixing up a real cop with one of his pals hired to scare off Barney from the pool), but Barney shows up at the window and gives Fred a piece of his own cake that he never got to eat. Fred is literally in tears from this token of kindness.
On top of that, Barney also bails him out of jail the next day (Wilma gave him the money, but it's the thought that counts).
It's small and short, but Bill and Joe's cameo at Pebbles and Bamm-Bamm's wedding in I Yabba-Dadda-Do.
The song that plays during the Flintstones and Rubbles taking care of the twins in Hollyrock-A-Bye-Baby.
The flashback scene to when Pebbles and Bamm-Bamm were introduced to each other.
Seeing the normally loud, loutish Fred trying his best to keep Wilma rested during her pregnancy is very sweet. This reaches its climax in "The Dress Rehearsal," when Wilma gives birth to Pebbles. The excitement and joy on Fred's face are adorable, especially when he rushes to the maternity ward window and mistakenly believes that all of the babies inside are his.
Ending of Dino Goes Hollyrock. Where Fred realizes his mistake and burns the money he received and puts out the candle with his own tear thinking he'll never see Dino again. And then Dino comes back and it's all the more reason for tissues.
During the lead up to Pebble's birth- when Fred and Wilma didn't know if they would have a boy or a girl- Fred buys a bunch of baseball equipment for the baby, on the hopes that "he" would be a potential baseball star. When Wilma points out that "junior" as Fred called it might be a girl, he thinks it over for a second, and then says that it'd be great, that she'd be the first female player to make the big leagues. It shows that Fred is just happy with the idea of being a father, and will be proud of his child no matter what.
Between his tiptoeing and how terrified of his wife's wrath he is, Fred giving his family savings to the Rubbles so they could adopt a child was really heartwarming. It's also what cools off Wilma, as Fred usually isn't capable of something so kind (or that complex for that matter).
Bamm-Bamm calling Barney "daddy".
Barney's poem to Fred at the bowling alley.
In The Flintstones in Viva Rock Vegas Wilma brings her friends to her father Colonel Slaghoople's birthday party. It's the first time they discover that she's an extremely wealthy heiress, and as such, the party is fraught with tension. When Wilma's alone for a while, her father enters with a box:
Colonel Slaghoople: I brought you a little something for your birthday!
Wilma: Daddy, it's your birthday.
Colonel Slaghoople: Oh! Well, these won't fit me...
(He opens the box to reveal a pearl necklace—Wilma's most iconic piece of jewelry.)
Wilma: Oh, Daddy...
Colonel Slaghoople: I know you don't like showy things, but...Wilma, dear, I want you to know that no matter what you do, your daddy will always love you, and couldn't be prouder of you.