The focus $F$ of the curve $C\_1$: $y^{2}=2px (p > 0)$ is $F(\frac{p}{2}, 0)$. Therefore, the value of $c$ for the hyperbola is $\frac{p}{2}$.

Since the line connecting the intersection points of the curves $C\_1$ and $C\_2$ passes through point $F$, by symmetry, this line is perpendicular to the $x$-axis. Let $x=c$, and substitute it into the equation of the hyperbola:

$$y^{2} = b^{2} \left( \frac{c^{2}}{a^{2}} - 1 \right) = \frac{b^{4}}{a^{2}}$$

Solving for $y$, we get $y = \pm \frac{b^{2}}{a}$, and thus the length of the chord $|MN| = \frac{2b^{2}}{a}$.

Now, let $x = \frac{p}{2}$, and substitute it into the equation of the parabola:

$$y^{2} = p^{2}$$

This gives us $y = \pm p$, and thus the length of the chord $|MN| = 2p$.

Equating the two expressions for $|MN|$, we get:

$$2p = \frac{2b^{2}}{a}$$

This simplifies to $b^{2} = 2ac = c^{2} - a^{2}$. Solving for the eccentricity $e$, we have:

$$e^{2} - 2e - 1 = 0$$

Solving this quadratic equation for $e$, we obtain $e = 1 + \sqrt{2}$.

Answer: $\boxed{D: \sqrt{2} + 1}$