Since PQ and RS are parallel lines, their slopes must be equal. The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

For the points P(1, q) and Q(3, 2), the slope of line PQ is:

\[m_{PQ} = \frac{2 - q}{3 - 1} = \frac{2 - q}{2}\]

For the points R(3, 4) and S(5, 2q), the slope of line RS is:

\[m_{RS} = \frac{2q - 4}{5 - 3} = \frac{2q - 4}{2}\]

Since PQ and RS are parallel, their slopes must be equal:

\[\frac{2 - q}{2} = \frac{2q - 4}{2}\]

Now, solve for q:

\[2 - q = 2q - 4\]

\[2 + 4 = 2q + q\]

\[6 = 3q\]

\[q = \frac{6}{3} = 2\]

So, the value of \(q\) is 2.

Therefore, the answer is:

D. 2