Draco Malfoy has lived his entire life feeling jealous of and inferior to Harry Potter. And he was right to: Harry was smarter, nobler, and had a way cooler scar. But Draco should feel comfortable knowing that he's better than Harry at one major thing: being a dad.
(Huh. Never saw that one coming.)
Draco's main goal in the first part of the play is to persuade the Ministry to "release a statement reaffirming that all Time-Turners were destroyed in the Battle of the Department of Mysteries" (1.4.77).
On the surface, it seems like Draco's doing this to protect his son. See, there's a rumor that Scorpius' dad is Moldy Voldy, and Draco wants to dispel these rumors. If the Ministry releases a statement, it will be harder for people to say that Draco's dead wife Astoria went back in time to bump uglies—and we do mean ugly—with Voldemort.
However, we learn that Draco might have an ulterior motive in the second half of the play. If the Ministry were to release a statement that all Time-Turners were destroyed, Draco would get himself off the hook. He's in possession of one, and the Ministry's statement would make it easier for him to keep it secret.
In this half of the play, though, Scorpius is missing. And, we never thought we'd say this, but Draco Malfoy and his son have one thing in common—they'll make a sacrifice for someone they love. Draco gives up his Time-Turner for the sole purpose of saving his son.
Yeah. Draco does something incredibly noble.
Working together with Harry, Draco is pretty much accepted by the team. Does that mean he can leave all his bitterness toward Harry Potter in the past, where it belongs? We sure hope so.
Draco also admits why he was so nasty as a child, saying,
DRACO: I was alone. And it sent me to a truly dark place. (2.15.12)
It's hard not to feel for him in that moment, and to see how hard he works to make sure that his son has a better childhood than he had. Scorpius might one day buy his dad a "#1 Dad" mug. Albus is more likely to take a mug and smash it over his own dad's head.