A Parallel from Symmetric Points

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with left vertex $A$ and upper vertex $B$. Let $P$ and $Q$ be two points on the ellipse symmetric about the origin $O$. The line $PA$ meets the $y$-axis at a point $M$, and the line $QA$ meets the line $y=\sqrt 3$ at a point $N$. Prove that $MN\parallel BP$.