Even in the first game there were hints of this. If free will is on and a parent is left doing nothing while their child is asleep, they will sometimes go tuck them in and kiss them goodnight. Awww! They do in the second game too.
Having an Elder Sim die of old age in The Sims 2 at gold or higher aspiration. They're greeted with zombie hula dancers, and Death greets them with a cocktail and tells them they're going on an eternal vacation.
In Sims 2: Pets, it's possible (but very hard) to befriend and adopt pet wolves.
In Sims 2, have a couple wake up in the morning without any task set up. The couple will proceed to cuddle in bed!
When a child is crying in Sims 2, you have the option to send an adult over to console them.
In some situations, this is automatic. For example, if a child witnesses their parent get into a fight with another adult (does not have to be the other parent), the child may get very upset and run off to their room in tears. After the fight or argument ends, the parent will go find the child and comfort them.
The cute picture in the Goth family album in Sims 2 of teen Cassandra playing with baby Alexander while their parents look on fondly.
Various patches occasionally add some pretty significant things beyond just fixes and minor feature changes/additions (stairs over gaps? Yay but unlikely to change your world). For example, one patch added in new animations... including a whole set of child/toddler animations thus allowing them to hug, have the child feed the toddler, and play peek-a-boo. D'awww...
Maybe it's just me, but even adults interacting with toddlers is cute to me. Like when an adult puts a toddler on the floor, and they give him/her a kiss.
In Pets, whenever a pet dies: The Grim Reaper shows he loves pets. For example, he hugs a dog or a horse when it dies and cuddles with the kittens.
Whenever a couple is sleeping together in the same bed, sometimes one will hug their spouse from the back. It's so sweet and heartwarming.
Reviving Erik Darling in The Sims 3 could be ruled as this.
The Generations expansion pack for The Sims 3 allows teens and older to read bedtime stories to sleepy children. After the child falls asleep, the older Sim will tuck the child in and give him or her a goodnight kiss.
The beginning of the intro cutscene in Sims 3 where a father is flipping pancakes and the daughter feels her pregnant mother's belly, then they hug.
In The Sims 3, when you have the pets expansion installed, and (as far as this troper has seen) have a single adult male or female and a pet, in the pets' relationships panel the owner will be shown as mother/father.
If you make a parent and child sim sit next to one another in the Sims 4, you can get them to cuddle. The animation that follows is absolutely adorable.
In a similar fashion, the sims can express love towards their child in ways that only a parent could understand; the parent hugs their child, the child is appreciative (if a little embarrassed/confused) followed by the parent pinching the child's cheek. Aaaw!
Both the reveal and launch trailers for The Sims 4 Cats & Dogs are heartwarming:
The reveal trailer has two moments in particular. One sequence has a Sim grow up with a family dog, only for him to get sick and implicitly die. While also a Tear Jerker, it's shown that her experience with him inspired her to love animals and become a vet. The other moment comes when we see a cat looking irritable throughout the video, but eventually becoming friends with his owner's date's dog. The next shot immediately after shows the owners' wedding... and both pets are there. For bonus points: both owners are male.
The official launch trailer opens with shots of a family making friends with a puppy they got on Christmas, a cat begging for pets, a dog jumping for joy at being fed, a dog waiting by his master's grave and being reunited with his ghost, and a stray cat being taken to the vet and being adopted. It emphasizes we (and, well, Sims) don't just need pets, but they need us as well.
Toddlers in The Sims 4 generally look like they came out of Pixar movies. They're utterly adorable, and older Sims will occasionally get positive moodlets from just how cute those toddlers are.
In The Sims 4 Get Famous, being a celebrity and interacting with child fans.
Many of the quests in Medieval are very, very silly... but I defy you not to go "Aaaaaw" should you decide to send your Peteran Priest off to soothe/convert the resident rabid chinchillas (as opposed to the "Kill their furry bahookies" approach). At the end of the quest, you get a cute graphic of some enlightened and reverent chinchillas along with this final line:
The docile chinchillas roamed out into the sunlight, calm and serene. Prior (Whatever-his-name-is) smiled at them and they - like old friends - smiled back.
You may melt now.
Any of the quests where the intent is to get the monarch married could count as well.
As well as any quest where there is a threat to the kingdom that allows you to peacefully remove the threat rather than outright kill it. Not to mention most of these "threats" had no intention of harming anyone in the first place.
The last "story" of this trailer, with the dog, the kid and the bullies.
Unless if you have many things to do, when a parent gets home and one of their children runs to greet them with a hug. It's a giant awww worthy moment.
Generally any interaction between a child and their parent or a child and their toddler sibling.
In the Pets expansions, if a Sim's pet loves them enough, they have a chance to save them from the Grim Reaper.