Aborted Arc: In season 4, Adam & Kristina lie to their daughter Haddie about a major medical issue. They claim Kristina is 100% cancer-free when she isn't. They do this, presumably, to convince her it's 100% fine for her to fly back across the country to the college she's attending instead of take a semester off to help out at home. This is clearly something that should bite them in the butt, especially if something goes drastically wrong. But when something does, and Kristina is in the hospital around Christmas and might die from a severely compromised immune system, Haddie shows up and simply hugs her mom. The lie and potential for Haddie to be either very angry and betrayed, or forgiving and understanding is 100% ignored/forgotten.
This is also done between Season 2 and Season 3 of the series. At the end of Season 2, Sarah writes a play, gets a big-name agent Zeek knows, and the play is a hit when performed locally. Maybe it would have flopped in bigger venues, and with the play seeming to be partially autobiographical, Sarah may have had no additional ideas. However, one would at least have expected a line in Season 3's premiere, "No producer wanted to pick it up for LA, much less New York, and I don't have any other ideas that seem worth it." Instead, Sarah went back to her (then) low-paying job as a bartender.
One episode has Lauren Graham and Mae Whitman's characters talking about a movie they'd just seen that stars Michael Cera. Funny, I almost forgot who played Michael Cera's girlfriend back on Arrested Development (and kick Cera's ass in Scott Pilgrim vs. The World). Mae Whitman's character also wear a banana costume for Halloween, which is a reference to what is worn by the employees on Arrested Development at their Frozen Banana stand.
Lauren Graham's character Sarah's first (albeit short-lived) love interest is a barista, and later on she dates her daughter's teacher, who is younger than her. In Graham's starring role in Gilmore Girls, her character was a coffee addict who eventually fell in love with the man who served her coffee (He was a diner owner and served her food also, but she came in just for coffee often), and she dated early on— and got engaged to— her daughter's young teacher.
In Season 3 Alex is explaining to Haddie why he doesn't think he should go to a high school party. Max says it's not a good idea because of the underage drinking, which he knows about because he watches Friday Night Lights. Max is telling this to Alex, played by Michael B. Jordan who not only was a star on Friday Night Lights for two seasons but who also got his role on Parenthood through Jason Katims' executive producer position at both shows.
Adults Are Useless: Some of the parents have this problem; Julia can't seem to do much of anything right with her daughter, Crosby is a Bumbling Dad for quite a few episodes. Adam and Kristina have mostly ignored their daughter while trying to deal with their son, who has Asperger's Syndrome.
The Alcoholic: Alex is in AA. Seth (Amber and Drew's father) is an alcoholic and has other substance abuse problems as well.
Amazingly Embarrassing Parents: Sarah. She yells after her kids to "make good choices!" on their first day of school. After this, Amber asks her to drop her off a few blocks from the front door. Adam and Kristina also qualify, considering their "MY DAUGHTER IS DATING!" freakout.
Asperger Syndrome: Adam and Kristina's son Max is diagnosed with it in the first episode. In Season 5, Hank fears he may have it too after getting a book on it from Adam, and starts seeing Max's specialist Dr. Pelican.
Bald Woman: Kristina for the last chunk of Season 4 as a result of getting chemotherapy.
Fragile Flower Kristina handles nothing well, though it's a 50/50 chance as to whether she'll cry or become comically irrational. Worse now that she's given birth to the latest Braverman.
Family Versus Career: Beautifully averted with Julia; her husband Joel stays home with their daughter while she goes to work as a successful lawyer. As of Season 4 though she's had to leave her job due to now having two kids to take care of.
Happily Married: Zeek and Camille, Adam and Kristina, Julia and Joel, at least until their marriage hits the rocks in Season 5.
Happy Ending: Season 4 had one. Kristina is cancer-free and gets to go to Hawaii with Adam, Jasmine is pregnant with her and Crosby's second child, Victor has become Happily Adopted and is no longer a Jerkass, Amber and Ryan are planning to get married, Drew gets to go to college, and Hank gets to go be with his daughter in Minnesota.
Hollywood Dateless: A female example with Sarah; early in the season it is a joke between the women that she hasn't had a date in years, despite being attractive. Possibly justified in that she has two teenagers, and all men are terrified of women who already have kids. Totally not justified in that later on, two very different men who know she has children fall for her very quickly.
I Have This Friend: Zeek asks Drew for advice about his marriage this way. Drew sees right through it.
Informed Flaw: Up until the point where he cheats on Jasmine, Crosby's irresponsible/deadbeat nature is this. Other Characters treat him as though he's a "screw up" and constantly talk him down, but the audience is only treated to a likeable character responding rather well to sudden fatherhood. It's not until after his infidelity that he starts actually exhibiting the informed behavior on screen.
The Runaway: Drew, Amber and Haddie run away from home once. Drew and Amber were easily found, while Haddie went to her grandparents' house.
Serious Business: The missing vending machine is this for Max, and in fact its what motivates him to become his school's president. Lucky him, the vending machine is ultimately brought back thanks to Kristina's influence.
Shout-Out: The scene in "Solace" where the TV automatically turns to Sponge Bob Square Pants might be a reference to a scene in Zathura where one of the brothers was watching sports and one of them changes the channel to Spongebob unexpectedly (Although, unlike Parenthood, the boy changed it during the middle of an episode, not the intro theme like the Parenthood episode).
The Slacker: Crosby is one, before Jasmine and Jabbar come into his life.
The Smart Guy: Max is quite knowledgeable in a variety of different things, per common Aspergers symptoms.
This Is Unforgivable: How everyone reacts when they learn of Crosby's cheating on Jasmine. They do all eventually forgive him (even Jasmine!) but it takes a while.
Throw the Dog a Bone: However badly Crosby suffers in his role as the quintessential Butt Monkey, this usually winds up happening at some point down the line.
After spending some time as The Woobie, Drew gets in to a good college at the end of the most recent season.
Took a Level in Jerkass: Victor takes quite a few levels in Jerkass in "Keep on Rowing" and "Small Victories". Ryan also takes a brief but nasty one when drunk that causes Amber to (at least temporarily) break up with him
Tropaholics Anonymous: Alex is in AA. Adam and Kristina attend a support group for parents of children with Asperger's.
What Could Have Been: Maura Tierney was originally cast as Sarah, but she developed breast cancer during production of the first season, which led to the role being recast with Lauren Graham and the series premiere being pushed back from September 2009 to March 2010.