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Tear Jerker: Leverage
Parker's fear for the orphans in "The Stork Job":
"You put these kids in the system, and odds are, they're gonna, they're gonna...they're gonna turn out like me."
Every moment Parker is onscreen during "The Stork Job" is either a Crowning Moment of some stripe or one of these.
"The Future Job": A fraudulent psychic uncovers a secret Parker has never told anyone: that she witnessed (and blames herself for) the death of her brother when they were children. Made worse by the fact that Parker has No Social Skills — she can't interpret normal nonverbal cues, and thus has a hard time understanding cold reading.
The story of her brother's death gets even worse when you remember she's also an orphan.
"The Maltese Falcon Job", The season two finale. After spiraling further and further out of control following the loss of Sophie, Nate finds himself at the mercy of his rival Sterling who wants to exchange a gun smuggler he's pursuing and Nate's team in exchange for Nate's freedom—because Nate is not like them. In the end, Nate captures the smuggler, then exchanges him for his teammates' freedom. While the team escapes, an FBI mook asks "Who is this guy?" It's Nate, bleeding and exhausted. For the last two years, he prided himself for being above his team. Now, he laughs and says, "I'm a thief."
In particular, Dorothy!Parker's face when Charlie!Hardison jumps on the train, leaving her forever.
Not to mention the fact that Dorothy had already died several years prior, meaning Charlie will never be reunited with her in life.
The flashback in "The Second David Job" where Nate watches his son Sam die (complete with a particularly harrowing use of the Big "NO!") certainly qualifies, especially since we as an audience know exactly what it did to him. The fact that the same flashback is shown several other times doesn't make it any less painful to watch.
Nate's Big "NO!" is especially heartbreaking as its more of a scream of pain than anything else.
Nate's whole conversation with Maggie, wherein we first see that flashback, where he confesses the truth about Sam's death is also pretty tearjerker-y.
"The Lost Heir Job" actually gave a tragic backstory to a character who in almost any other episode would have been the Villain of the Week. Corrupt Corporate Executive falls for the stripper he got pregnant and plans to marry her. His lawyer who cleans up after him doesn't want the scandal and sends her off to Las Vegas, where she has their daughter and dies of cancer two years later. The guy manages to track his now Happily Adopted daughter down and begins to gradually build up a relationship with her again through her charity. But then his lawyer cuts off contact with anyone and he never gets to explain anything to his daughter. The last time he sees his daughter he mistakes her for her mother who he thought had abandoned him and is thrilled to see her.
Sophie and the crew invoke this in the climax of "The San Lorenzo Job" to turn the people of the titular country against corrupt President Ribera.
Two for the price of one episode in the season 4 premiere, "The Long Way Down Job": Parker's pleas of wanting to bring the body of the dead husband of their client, and the victim's goodbye message to his wife from his phone.
Parker: You mean... he died down here? ...Alone?
Season 4 in general is filled with tearjerkers galore.
Especially with the second-to-last episode of the season: "The Radio Job". Jimmy Ford makes a deal with Latimer, despite his son's warnings, because it was all to keep him safe (and for two million dollars). He gets knocked unconscious and wakes up slowly by Nate's phone calling him, and realizes he's surrounded by explosives and a timer just about to go off. Jimmy calmly tells Nate to stay away, who keep telling him he's going to get him, losing his cool and rising his voice to the point of fear because he knows what's about to happen. He gets out of the car and hears his dad say he loves him... forcing him to a shocked stupor. Then the building blows up, all in slow motion.
Jimmy: Tell them stories about me. Tell them...tell them how much Jimmy Ford loves his son.
"The Grave Danger Job": Parker talking Hardison through being Buried Alive.
The entire plot of "The D.B. Cooper Job". Over Nate's objections, the team helps McSweeten's dying father, an FBI agent, with the one case he could never solve—the Real LifeD.B. Cooper skyjacking. It turns out that Cooper was actually Steve Reynolds—the elder McSweeten's own partner, whom he himself recruited into the FBI. While McSweeten never caught "Cooper", he did something even better by turning him into a good person. While this plot is heartwarming, it also belongs here because it's possible that the elder McSweeten dies without realizing the truth. Some fans theorize that he figures it out, but the episode doesn't say.
"The Broken Wing Job", when Parker goes through the voice mail of the doctor who orders Chicken Parmesan every day. The thief thinks she can crack some jokes at the guy for always eating one bite before returning it, but then hears some unsettling news...
27 days ago: Hi James? It's Larry, at the hospital. We, uh... just heard about Emma. My god, I'm so sorry. We're all so sorry.
11 days ago: Hey James, it's me, uh... just checking in. If there's anything you need, just call, OK? The service was beautiful.
Today: I just want to let you know we've shifted your surgeries to Doctor Phillips. But we could really use you back here, buddy.
At the end of "The Low Low Price Job", Eliot returns to his childhood home to try and make amends with his father, whom he had left on bad terms with years prior. However, nobody answers the door, and the look of crushing defeat on Eliot's face as he walks away is heartrending. It's not clear if his father refused to answer the door, or had already passed away in the time they were apart.
At the end of "The Toy Job", Nate's story to the team about a trumpet he had as a child. He eventually gave it to Sam—who got sick and died before he could learn to play it. Eliot and Sophie both looked like they were about to cry.
Elaboration later due to spoilers, but THE FINALE.
It starts off bad, then the emotional baggage just piles on and on. Then you hit the bridge scene, where Nate's driving the truck, and he looks back. It's heart-wrenching.