To solve this problem, let's go through the following steps:

Step 1: Identify the slope of the given line

The line equation given is: $x + 3y = 6$ To find the slope, we should convert it to slope-intercept form $y = mx + b$, where $m$ is the slope.

1. Rearrange for $y$: $3y = -x + 6$
2. Divide by 3: $y = -\frac{1}{3}x + 2$

The slope of this line is $m = -\frac{1}{3}$.

Step 2: Find the equation of a line parallel to this one

Since parallel lines have the same slope, the line we are looking for will also have a slope of $m = -\frac{1}{3}$.

Step 3: Use the point-slope form to find the equation of the new line

We are given a point $(-3, 2)$ through which the line passes. Using the point-slope form: $y - y_1 = m(x - x_1)$ Substitute $m = -\frac{1}{3}$, $x_1 = -3$, and $y_1 = 2$: $y - 2 = -\frac{1}{3}(x + 3)$ Simplify this equation:

1. Distribute $-\frac{1}{3}$: $y - 2 = -\frac{1}{3}x - 1$
2. Add 2 to both sides: $y = -\frac{1}{3}x + 1$

Step 4: Identify the correct answer

The equation we derived is: $y = -\frac{1}{3}x + 1$

Looking at the options, the correct answer is: $\boxed{y = -\frac{1}{3}x + 1}$

Would you like further details on this solution, or do you have any additional questions?

Here are five related questions to explore further:

1. What is the formula for determining if two lines are parallel?
2. How would you find the perpendicular line to $x + 3y = 6$ through a given point?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the significance of the slope in different forms of line equations?
5. How can you verify if two lines are identical or just parallel?

Tip: When dealing with line equations, always rearrange them into slope-intercept form ($y = mx + b$) to easily identify the slope.