If we have two points $ (x_1,y_1) $ and $ (x_2,y_2) $ on a line, we can find the slope of the line by using the formula $ \dfrac{y_1-y_2}{x_1-x_2} $. So, for the line we are given, the slope is $ \dfrac{(-5)-(-14)}{2-p}=\dfrac{9}{2-p}, $ and the slope is also $ \dfrac{(-14)-(-17)}{p-(p+2)}=\dfrac{3}{-2} $. Setting these values equal, we get $$\dfrac{9}{2-p}=-\dfrac{3}{2}.$$ Multiplying both sides by the product of the denominators and simplifying gives \begin{align*} (2-p)(3)&=(-2)(9)\\ 6-3p&=-18 \\ p&=8.\end{align*} Now we need to find $ q $. Using the same strategy as above, we find that \begin{align*} \frac{q-(-5)}{13-2}&=\frac{3}{-2} \\ (11)(3)&=(-2)(q+5)\\ 33&=-2q-10 \\ q&=-21.5.\\ \end{align*} Therefore, $ p+q=8+(-21.5)=\boxed{-13.5} $.