Step-by-step Solution:

1. Identify the coordinates of P and Q. The point P is given as $P(3, 2, 6)$. Its position vector is $\vec{p} = 3\widehat i + 2\widehat j + 6\widehat k$. The point Q lies on the line $\vec{r} = (\widehat i - \widehat j + 2\widehat k) + \mu ( - 3\widehat i + \widehat j + 5\widehat k)$. The position vector of any point Q on this line can be written as: $\vec{q} = (1 - 3\mu)\widehat i + (-1 + \mu)\widehat j + (2 + 5\mu)\widehat k$ So, the coordinates of Q are $(1 - 3\mu, -1 + \mu, 2 + 5\mu)$.

2. Determine the vector $\overrightarrow{PQ}$. The vector $\overrightarrow{PQ}$ is found by subtracting the position vector of P from the position vector of Q ($\,\overrightarrow{PQ} = \vec{q} - \vec{p}\,$). $\overrightarrow{PQ} = [(1 - 3\mu) - 3]\widehat i + [(-1 + \mu) - 2]\widehat j + [(2 + 5\mu) - 6]\widehat k$ $\overrightarrow{PQ} = (-2 - 3\mu)\widehat i + (\mu - 3)\widehat j + (5\mu - 4)\widehat k$

3. Identify the normal vector to the plane. The equation of the plane is given as $x - 4y + 3z = 1$. The normal vector to this plane, $\vec{n}$, is determined by the coefficients of x, y, and z. $\vec{n} = 1\widehat i - 4\widehat j + 3\widehat k$

4. Apply the condition for parallelism. For the vector $\overrightarrow{PQ}$ to be parallel to the plane, it must be perpendicular to the normal vector of the plane, $\vec{n}$. The condition for two vectors being perpendicular is that their dot product is zero. $\overrightarrow{PQ} \cdot \vec{n} = 0$

5. Calculate the dot product and solve for $\mu$. Substitute the expressions for $\overrightarrow{PQ}$ and $\vec{n}$ into the dot product equation: $[(-2 - 3\mu)\widehat i + (\mu - 3)\widehat j + (5\mu - 4)\widehat k] \cdot [1\widehat i - 4\widehat j + 3\widehat k] = 0$ $(-2 - 3\mu)(1) + (\mu - 3)(-4) + (5\mu - 4)(3) = 0$ Expand the terms: $-2 - 3\mu - 4\mu + 12 + 15\mu - 12 = 0$ Combine the terms with $\mu$ and the constant terms: $(-3 - 4 + 15)\mu + (-2 + 12 - 12) = 0$ $8\mu - 2 = 0$ $8\mu = 2$ $\mu = \frac{2}{8} = \frac{1}{4}$

6. Conclusion. The value of $\mu$ for which the vector $\overrightarrow{PQ}$ is parallel to the given plane is $\frac{1}{4}$. This corresponds to option A.