If $P(1,2)$, $Q(4,6)$, $R(5,7)$, and $S(a,b)$ are the vertices of a parallelogram $PQRS$, then the values of $a$ and $b$ are:

In a parallelogram $PQRS$, the diagonals $PR$ and $QS$ bisect each other, meaning they share the exact same midpoint.
Using the midpoint formula $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$:

$$\text{Midpoint of }PR=\left(\frac{1+5}{2},\frac{2+7}{2}\right)=\left(3,\frac{9}{2}\right)$$

$$\text{Midpoint of }QS=\left(\frac{4+a}{2},\frac{6+b}{2}\right)$$

Equating the corresponding coordinates:

$$\frac{4+a}{2}=3⟹4+a=6⟹a=2$$

$$\frac{6+b}{2}=\frac{9}{2}⟹6+b=9⟹b=3$$

Thus, $a=2$ and $b=3$.