One arc has the family return from a trip only to find that their house has been broken into. Calvin is pacified immediately after finding Hobbes, whom they had accidentally left behind when they went on the trip. His parents, however, are notably shaken, and the realization that they aren't as safe as they thought they were lingers with them for the rest of the arc. Calvin's dad in particular has to come to terms with the fact that being a parent doesn't automatically equip you to deal with any problem, contrary to what he thought after idolizing his father when he was younger. Parents are people too, and what makes them seem invincible is the fact that they put on a brave face for the sake of their children, which he learns to do.
There's the story arc in which Calvin finds the dying raccoon. He brings his mother to help him save it, telling Hobbes: "You don't get to be Mom if you can't fix everything just right.". His mom admits though that there really is very little they can do to save the raccoon and it inevitably dies. This brings up the fact that parents can't always save the day and aren't always going to be able to protect their children from experiencing loss and death.
Adult fears are also treated humorously with Calvin using them as ideas for his Halloween costumes: a barrel of toxic waste, and nothing (just a child; think of what he and his generation receiving questionable influences will have grown up into when the adults he's trick-or-treating are old and weak).
In one series of strips, Calvin wanted the garage to himself, so he pushed the car out. It accidentally kept going and crossed the road, crashing into a ditch. Calvin, scared of being punished, ran away into the forest. His mom caught up with him, and while Calvin didn't really understand, any parent reading the strip definitely will—his mom wasn't mad, she was just terrified that he had been hurt. He didn't get in trouble because she was just glad he was okay.
When Calvin gets sick, his mother mentions that he won't be missing any school since it's the weekend. Calvin responds "I know" without making any fuss...and his mother runs to the phone to call the doctor. It turns out Calvin just "caught the bug going around" and will be fine in a few days. But given how weak and miserable he looked, Mom's fear was completely understandable.
Several times in FoxTrot. One arc had Roger coming home from work to find Andy and Jason gone. Paige tells him they're at the hospital, and that Jason was hit by a car. She meant to say it was a toy car (Jason had gotten hit on the chin with one and needed stitches), but Roger doesn't know that and promptly tears outta there to see Jason at the hospital. Then there's the arc where Peter goads Jason into going onto the roof, Jason loses footing and falls off, hitting his head and having to go to the hospital for supervision; Paige and Jason finding a needle at the beach (they throw it out, which freaks Andy out because she fears they accidentally pricked themselves when they did); Paige going to the dance with a lecherous date who clearly wants to have his way with her...
Another time, Paige babysits Ms. O'Mally's preschool daughter Katie, then falls asleep watching her. Katie starts playing with scissors. Paige manages to wake up in panic and saves Katie from hurting herself, but is at a loss to explain what happened to Katie's understandably angry mother. (Exactly why she continued to hire Paige after that boggles the mind; the only punishment Paige seemed to get was having to pay for Katie's dress that she ruined.)
Peanuts: Charlie Brown dealt with quite a lot of adult fears for an eight-year-old — one arc in particular had him lying, alone, in a hospital bed worrying that he was going to die and that the doctors weren't even going to tell him.
The cartoonist Quino, uses it in several strips, like this one◊. For those who don't understand Spanish: It's about a teenager who rebels against having to study Greek Mythology by saying that "he has nothing to do with it", and instead asks for permission to drive his father's car to a party, his father reluctantly agrees and when he leaves... his father starts reading the studies book and the story of Phaėton. The man's face in the third-to-last panel says it all.