Example 3 Given the ellipse $\frac{x^{2}}{2}+y^{2}=1$, find the equation of the locus of the midpoints of parallel chords with a slope of 3.
Let the endpoints of the chord be $M\left(x_{1}, y_{1}\right), N\left(x_{2}, y_{2}\right)$, and the midpoint of $M N$ be $R(x, y)$. Then, $x_{1}^{2}+2 y_{1}^{2}=2, x_{2}^{2}+2 y_{2}^{2}=2$,
subtracting the two equations and dividing by $\left(x_{2}-x_{1}\right)$, we get $x_{2}+x_{1}+2\left(y_{2}+y_{1}\right) \cdot \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=0$.
Since $x_{1}+x_{2}=2 x, y_{1}+y_{2}=2 y$, we have $x+2 y \cdot \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=0$.
Noting that $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=3$, the equation of the desired locus is $x+6 y=0$ (inside the ellipse).