Analysis

This problem primarily tests your understanding of the standard equation of a hyperbola and the application of its basic properties. Utilizing the standard equation of the hyperbola, we can derive the equations of its asymptotes and compare them with the given conditions to find the answer. This is considered a basic-level problem.

Step-by-step Solution

1. The standard equation of the hyperbola is given by $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{9} = 1$.
2. From the standard equation, we can derive the equations of the asymptotes, given by $y = \pm \frac{3}{a}x$ or $3x \pm ay = 0$.
3. Comparing the derived asymptote equations with the given asymptote equations ($3x \pm 2y = 0$), we can deduce that $a = 2$.

Therefore, select option A: $\boxed{2}$.