Family Guy has some. Most of which are parodies, and some of which are serious.
The simply titled "Brian & Stewie" (the 150th episode, the only episode in which Seth MacFarlane is credited for doing all the voicework, and the only episode in which there are no background music, cutaway jokes, flashbacks, or Take Thats against whoever's famous) where Brian and Stewie are locked in a bank vault for the entire weekend and Stewie discovers that Brian has a gun in case he wants to commit suicide. Sure, there was that really gross poop-eating scene and some musical numbers at the end (which were only in there in the first place because FOX had some gimmicky music-based "Fox Rocks" episode line-up on on their network for a week;note This also resulted in The Simpsons episode "To Surveil with Love" having that Couch Gag with everyone in Springfield lip-syncing to Kesha's TiK ToK and the Fringe episode "Brown Betty" being a Musical Episode. in all reruns and on the DVD, the musical numbers aren't there), but other than that, the episode was a serious look at the relationship between Brian and Stewie.
Another episode ends with this parody of what's found at the end of "very special episodes": "To learn more about drugs, go to your local library. There's probably someone behind it who sells drugs."
Parodied in the episode where Happy-Go-Lucky Toysnote The toy factory Peter used to work at before his boss died is merged with a cigarette company who uses the place as a front to get kids hooked on tobacco. First, it's played straight, with Peter enjoying the perks of being a company head and backing up his superiors — until Stewie starts smoking cigarettes, then Peter protests against them. In the end, the message was really about how it's wrong to kill strippers (a Call-Back to a short scene where a Senator is freaking out over killing one of the dancers at a D.C. strip joint called "The Oval Orifice" and Peter calming him down by saying that, yes, choking her to death on dollar bills and beating her with a chair were horrible ways for her to go, but at least she didn't die of cigarette smoking), because strippers were put on Earth to entertain horny men and most of them are already dead on the inside from the life decisions that led them to this point, so physically killing them is redundant.
"Screams of Silence: The Story of Brenda Q.", aside from the traditional Black Comedy moments, was actually a very serious episode that dealt with Quagmire's sister being abused by her boyfriend (which was first mentioned in "Jerome is the New Black"note The episode featuring Quagmire calling Brian out on being a pompous asshole who pushes his atheist and liberal views on others), and is one of the few (if not only) times that Domestic Abuse is actually shown in a serious light on the show.
Earlier in the series, there was an episode where the Griffin family meet a family of nudists (Peter saved the father from being swept in an undercurrent while fishing, thinking he lost his trunks in the water). Meg ends up dating the son, and the family aren't too comfortable with that. They eventually warm up to the idea, even exhibiting their own tolerance for the practice by going nude when he comes over. Despite the ruthless amount of innuendo and situation gags, it's really quite touching.
"Brian Wallows and Peter Swallows" had some funny moments, but was really a very special episode (and one of the few episodes, according to Seth MacFarlane and Alex Borstein, that actually has human emotion in it) about coping with loss (Brian caring for the elderly shut-in who used to be a jingle singer and Peter caring for the baby birds that have nested in his newly-grown beard).
Parodied at the end of an early episode in which The Griffins go to an Indian casino and Peter and Chris go out in the woods on a spirit quest. After Stewie makes a racist comment about Native Americans, Lois launches into a "The More You Know" PSA about how Native Americans are people too (followed by Stewie stating the same thing about Mexicans, Meg saying the same thing about Swedish people, and Peter bashing Canadians, and concluding the episode with "Canada sucks!")
"Livin' on a Prayer" was about Lois kidnapping a baby so it can be given cancer treatment much to the disapproval of his Christian Scientist parents, who don't believe in letting doctors treat illnesses.
The banned episode "Partial Terms of Endearment" was actually praised for showing realistic arguments for and against abortion — and, at least until the end where Peter flat-out tells the audience that the baby Lois was carrying was aborted, treated the subject matter with respect. No wonder it was banned from American TV (it is available on DVD and UK viewers have seen it on BBC3).
