1. Equation of the Ellipse: Let the equation of the ellipse be given by:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
where $a$ is the semi-major axis and $b$ is the semi-minor axis. The relationship between $a$, $b$, and the focal distance $c$ is given by:
$$c^2 = a^2 - b^2$$

2. Coordinates of the Focus: Let $F(\epsilon c, 0)$ be one of the foci of the ellipse, where $\epsilon = \pm 1$.

3. Point on the Ellipse: Let $P(au, bv)$ be a point on the ellipse, where $u^2 + v^2 = 1$.

4. **Slope of the Tangent at $P$**: The slope of the tangent to the ellipse at $P$ is given by:
$$m_{PP} = -\frac{b^2 x}{a^2 y} = -\frac{bu}{av}$$

5. **Equation of the Chord $CD$**: Since $CD$ is parallel to the tangent at $P$, its slope is also $-\frac{bu}{av}$. Therefore, the equation of $CD$ passing through the center $O(0,0)$ is:
$$y = -\frac{bu}{av} x$$

6. **Equation of the Line $PF$**: The line $PF$ passes through $P(au, bv)$ and $F(\epsilon c, 0)$. The slope of $PF$ is:
$$m_{PF} = \frac{bv - 0}{au - \epsilon c} = \frac{bv}{au - \epsilon c}$$
Thus, the equation of $PF$ is:
$$y - bv = \frac{bv}{au - \epsilon c} (x - au)$$

7. **Intersection Point $Q$**: To find the coordinates of $Q$, we solve the equations of $CD$ and $PF$:
$$y = -\frac{bu}{av} x$$
$$y = \frac{bv}{au - \epsilon c} (x - \epsilon c)$$
Equating the two expressions for $y$:
$$-\frac{bu}{av} x = \frac{bv}{au - \epsilon c} (x - \epsilon c)$$
Solving for $x$:
$$-\frac{bu}{av} x (au - \epsilon c) = bv (x - \epsilon c)$$
$$-bu x = bv (x - \epsilon c)$$
$$-bu x = bv x - bv \epsilon c$$
$$x (bv + bu) = bv \epsilon c$$
$$x = \frac{bv \epsilon c}{bv + bu} = \frac{\epsilon c v}{1 + \frac{u}{v}}$$
$$x = \frac{\epsilon c v^2}{v + u}$$
Substituting $x$ back into the equation of $CD$:
$$y = -\frac{bu}{av} \cdot \frac{\epsilon c v^2}{v + u}$$
$$y = -\frac{\epsilon b c u v^2}{a (v + u)}$$

8. **Length $PQ$**: The coordinates of $Q$ are:
$$Q \left( \frac{\epsilon c v^2}{v + u}, -\frac{\epsilon b c u v^2}{a (v + u)} \right)$$
The distance $PQ$ is:
$$PQ = \sqrt{ \left( au - \frac{\epsilon c v^2}{v + u} \right)^2 + \left( bv + \frac{\epsilon b c u v^2}{a (v + u)} \right)^2 }$$
Simplifying the expression inside the square root:
$$PQ = \sqrt{ \left( \frac{a (v + u) au - \epsilon c v^2}{v + u} \right)^2 + \left( \frac{a (v + u) bv + \epsilon b c u v^2}{a (v + u)} \right)^2 }$$
$$PQ = \sqrt{ \left( \frac{a^2 u (v + u) - \epsilon c v^2}{v + u} \right)^2 + \left( \frac{a b v (v + u) + \epsilon b c u v^2}{a (v + u)} \right)^2 }$$
$$PQ = \sqrt{ \left( a^2 u - \epsilon c v^2 \right)^2 + \left( b v \right)^2 }$$
$$PQ = \sqrt{ a^2 u^2 + b^2 v^2 + c^2 - 2 \epsilon a c u }$$
$$PQ = \sqrt{ a^2 - 2 \epsilon a c u + c^2 }$$
$$PQ = a$$

Thus, $PQ = OA$.

The final answer is $\boxed{ PQ = OA }$.