A Parallel Chord Swap

The ellipse $\dfrac{x^2}{4} + y^2 = 1$ has left vertex $A$. Points $P$ and $Q$ on the ellipse are symmetric about the $y$-axis, $O$ is the origin, and $B$ is a point on the ellipse with $AB \parallel OP$ and $B \neq Q$. Prove that $BQ \parallel AP$.