To find the value of $p$ that makes the two lines parallel, we need to compare the slopes of the two lines.

The general form of the equation of a straight line is $Ax + By + C = 0$, and the slope of the line can be found using the formula:

$m = -\frac{A}{B}$

Step 1: Find the slope of the first line $l_1$.

The equation of the first line is:

$l_1 : px - 3y + 5 = 0$

Here, $A = p$, $B = -3$, and $C = 5$.

The slope $m_1$ of this line is:

$m_1 = -\frac{A}{B} = -\frac{p}{-3} = \frac{p}{3}$

Step 2: Find the slope of the second line $l_2$.

The equation of the second line is:

$l_2 : 2x - 3y - 6 = 0$

Here, $A = 2$, $B = -3$, and $C = -6$.

The slope $m_2$ of this line is:

$m_2 = -\frac{A}{B} = -\frac{2}{-3} = \frac{2}{3}$

Step 3: Set the slopes equal.

Since the two lines are parallel, their slopes must be equal. Therefore:

$\frac{p}{3} = \frac{2}{3}$

Step 4: Solve for $p$.

By equating the slopes:

$p = 2$

Final Answer:

The value of $p$ that makes the two lines parallel is $\boxed{2}$.

Would you like more details or explanations on any part of the solution? Here are some related questions:

1. How do we derive the slope-intercept form of a line from its general equation?
2. What are the conditions for two lines to be perpendicular?
3. How can we find the distance between two parallel lines?
4. What happens to the slope of a line if the value of $p$ is negative?
5. How do we determine if two lines are coincident (i.e., the same line)?

Tip: When comparing the slopes of lines to determine if they are parallel or perpendicular, remember that parallel lines have equal slopes and perpendicular lines have slopes that are negative reciprocals of each other.