Consider each statement:

A. If $a \perp \alpha$ and $a \perp \beta$, it means that line $a$ is perpendicular to both planes $\alpha$ and $\beta$. According to the geometric property that if a line is perpendicular to two planes, then these two planes are parallel to each other, we can conclude that $\alpha \parallel \beta$. Therefore, option A is correct.

B. If $a \perp \gamma$ and $\beta \perp \gamma$, this means that line $a$ is perpendicular to plane $\gamma$ and plane $\beta$ is also perpendicular to plane $\gamma$. While both $\alpha$ and $\beta$ are perpendicular to $\gamma$, this does not necessarily imply that $\alpha$ and $\beta$ are parallel; they could intersect. Hence, option B does not guarantee that $\alpha \parallel \beta$.

C. If $a \subset \alpha$, $b \subset \beta$, and $a \parallel b$, this provides information about lines being parallel within the respective planes. However, this does not ensure that the planes themselves are parallel. They could intersect as well. Thus, option C does not suffice to deduce that $\alpha \parallel \beta$.

D. If $a \subset \alpha$, $b \subset \alpha$, $a \parallel \beta$, and $b \parallel \beta$, it seems to suggest that the lines in plane $\alpha$ are parallel to plane $\beta$. However, this is not sufficient to conclude that $\alpha$ and $\beta$ are parallel because $a$ and $b$, while inside $\alpha$, could be skews or intersecting lines, and their relationship to $\beta$ doesn't define the relationship between the planes. Therefore, option D is not sufficient either.

The correct option is A:

$$\boxed{\alpha \parallel \beta \text{ is concluded from option A}}$$