This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / The Cosby Show
"It seems like just yesterday I was holding my baby in my arms. And today my baby had two babies. Now she's a mother..."
One-shot character Ray Palamino, played by Sammy Davis Jr., introducing himself to his pregnant granddaughter who he hadn't seen in over 20 years.
"Hey everybody! My name is Ray Palamino. I'm 67 years old... and I can't read." As much as the character claims that it doesn't bother him, the strain in Sammy Davis Jr's voice completely contradicts it.
Also Clair's speech to Theo why she's so proud of him graduating from college in the final episode.
Going back to season 2, there's the letter that Cliff and Clair read to Denise after she announces her college decision.
A positive example: Cliff's friend Jake and Jake's friend Stanley finally resolving a feud they'd had between each other for a decade after learning their children were getting married to each other and were prepared to have the wedding without their fathers present if it became necessary.
Cliff moping over the old couch and chair that is to be replaced. It's played for laughs and seems a little ridiculous, but it can really strike a chord with anyone who's ever had a sentimental attachment to a piece of furniture.
Clair going up to check on Olivia, opening the door, and seeing Denise holding her and reading a book to her, and closing the door. As overbearing as Clair can be sometimes, she knows when to back off.
When Denise meets Martin's first wife and Olivia's mother, Paula, Paula tells the story of how her career path, getting married, and having a child were because of her parents' wishes rather than her own. For her it got to the point of suffocation and she realized she had to get out, but that doesn't stop her from admitting how painful the decision was to her and how much it hurt Martin and Olivia. On the plus side, it also reveals that she and Martin still get along very well and she and Denise become very good friends even to the point of her helping Denise finally figure out what she wants to do in life.
"Off To See The Wretched": Oh yeah. Cliff and Clair were pissed because they assumed their daughter was dead. And Vanessa going to bed in tears after getting thoroughly chewed out by Clair.
The November 19, 2017 death of Earle Hyman (Grandpa Huxtable).