The line through $ \mathbf{a} - 2 \mathbf{b} + 3 \mathbf{c} $ and $ 2 \mathbf{a} + 3 \mathbf{b} - 4 \mathbf{c} $ can be parameterized by \begin{align*} &\mathbf{a} - 2 \mathbf{b} + 3 \mathbf{c} + t((2 \mathbf{a} + 3 \mathbf{b} - 4 \mathbf{c}) - (\mathbf{a} - 2 \mathbf{b} + 3 \mathbf{c})) \\ &= (1 + t) \mathbf{a} + (-2 + 5t) \mathbf{b} + (3 - 7t) \mathbf{c}.\end{align*}To get an expression of the form $ p \mathbf{b} + q \mathbf{c}, $ we want the coefficient of $ \mathbf{a} $ to be 0. Thus, we take $ t = -1, $ which gives us $ -7 \mathbf{b} + 10 \mathbf{c} $. Hence, $ (p,q) = \boxed{(-7,10)} $.