How does Sir Gawain's quest parallel the quest of all Christians as they move through life?
I'm not sure one can say that Gawain's quest parallels the lives of all Christians. There are well over a billion Christians alive in the world today and their lives and faith journeys cannot all be assumed to be identical.
In many ways the tale of Sir Gawain and the Green Knight is one of coming of age of a brash young knight who learns that boastfulness and military prowess are less important than humility and ethical conduct. In his initial encounter with the Green Knight, Gawain demonstrates physical strength, an attribute of youth, courage, and loyalty to Arthur, but also a degree of pride and longing for glory. The figure of the Green Knight, who survives decapitation, is Christlike.
When Gawain arrives at the castle, he is subject to temptations. In so far as he has learned the self-control necessary to resist the temptation of adultery, he is rewarded, but his lack of faith, demonstrated by his taking the sash causes him to be chastised by the Green Knight who serves the role of Christ the Teacher, an image the author would have known, inter alia, from St. Augustine's De Magistro. By seeing the positive results of moral behavior and the negative results of cowardice, lack of faith, and hypocrisy, Gawain learns to be a better person.
One could argue that many Christians have similar experiences in there lives, being rewarded for moral behavior and learning moral lessons from that, but many Christian theologians would say that the Christian God does not reward good behavior in this life, turning morality into something that "buys" good fortune and even salvation, but rather that good works should spring from faith and gratitude for the sacrifice of Christ, not from desire for earthly rewards or glory.