Parker is the chief engineer aboard the Nostromo, and he's the type of guy who will tell you how he feels—and then back it up with his fists. Even before the alien shows up, we can tell that Parker is a man who won't shy away from confrontation, whether it's with his crewmates or a freaky alien visitor.
In the opening scene, Parker can hardly wait for the breakfast banter to die down before bringing up the totally unfair bonus situation. Brett may be the one who reminds Parker to bring up the situation, but Parker actually engages Dallas in the argument while Brett follows up with his passive, "Right."
Got it: we know who the alpha male is in this duo.
Dallas tells Parker, "You get what you're contracted for just like everybody else" (Alien), but Parker can't let it go. He brings it up again when Ripley comes down to check on the repairs, and shows a similar persistence when, say, deciding whether to abandon his crew-mates. When Kane arrives on the scene with his face-loving pal, Parker immediately goes to the nuclear option: freezing him. After they lift off from the planet, Parker brings it up again, telling Dallas, "I think we should freeze him. If he's got a disease, stop it where it is" (ibid).
Way to escalate things, Parker. Even Ripley just wanted to quarantine him for a few hours.
Once the alien starts chowing down on his crew-mates, Parker drops the whole bonus thing, but his aggressive attitude sticks around. If anything, he takes it to the next level once he's got an actual enemy to fight.
Man of Action
As the alien becomes a clear(er) threat to the crew, Parker's nature as a man of action really shines. Forethought? Strategy? Not this guy. When planning the alien hunt, Dallas even has to remind him, "Parker, I don't want any heroics out of you, all right?" (ibid).
After Dallas's death, Parker just can't even with all the talking and thinking. The crew gathers to think about what to do next, and Parker exclaims, "How come nobody's saying nothing around this place?" As if that weren't enough to tell us that he's ready to get this done, he grabs the flamethrower and starts checking it, as though he's itching to use it. And then when Lambert talks about drawing straws to determine who will go on the shuttle, Parker says, "I'm not for drawing straws. I'm for killing that god damn thing right now" (ibid).
Annoying, maybe. But Parker's act-first-think-later (if you're not dead) does have its benefits. Parker is the one who knocks out Ash, revealing him to be a robot; Parker is the one who launches himself at the creature attacking Lambert when he can't get a clear shot.
Oh, sure, the heroic action just ends up getting them both dead, but still. It's the thought (and deed) that counts, right?
Foil Me Once, Shame on You
Parker, Lambert, and Ripley are the final three survivors for a good reason: all three play foils to one another. Parker's action-man mentality provides a direct opposite for Lambert's passive, emotional nature, and with Ripley somewhere in the middle: sensitive, but with a kick-butt core.
Parker-the-action-man seems to be okay with Lambert, because he can play protecting hero to her. Ripley, though, gets on his nerves a bit: not weak enough to need him, but not strong enough (in his eyes) to lead the ship. When Ripley says she's coming down to engineering to check on his work, he bellyaches, "I'd like to see what she'll do" (ibid). (Notice the "do": again, for Parker, all that matters is action.) Later, he pushes her action enough that she shouts, "Listen to me, Parker! Shut up!"
By the end of the film, these two do eventually see eye-to-eye. Literally. After Ash is killed and Ripley decides to blow up the ship. Parker responds, "Good." In this scene, the camera frames the two characters dead center, suggesting that they've come together and set aside their differences. Parker even tells her to take care of herself when she goes to prep the rocket, something earlier Parker wouldn't have done.
While Parker never learns to think before he acts, he eventually learns to respect Ripley and her role as a leader. Unfortunately, he only has a short time to bask in his character growth before becoming alien chow.