Given that the two sides of angle $\alpha$ and angle $\beta$ are parallel, we know that angle $\alpha$ and angle $\beta$ are either equal or supplementary. This is a key property of parallel lines intersected by a transversal.

Let's denote the measure of angle $\beta$ as $x^\circ$. According to the problem, angle $\alpha$ is $36^\circ$ less than three times angle $\beta$. Therefore, we can express angle $\alpha$ as $3x^\circ - 36^\circ$.

### Case 1: Angle $\alpha$ and angle $\beta$ are Equal
If angle $\alpha$ and angle $\beta$ are equal, we have:
$$x = 3x - 36$$
Solving for $x$, we rearrange the equation:
$$2x = 36$$
$$x = 18$$
Thus, if angle $\alpha$ and angle $\beta$ are equal, $\angle \beta = 18^\circ$. Substituting $x = 18$ into the expression for $\angle \alpha$, we get:
$$\angle \alpha = 3 \times 18^\circ - 36^\circ = 54^\circ - 36^\circ = 18^\circ$$

### Case 2: Angle $\alpha$ and angle $\beta$ are Supplementary
If angle $\alpha$ and angle $\beta$ are supplementary, their sum equals $180^\circ$. Therefore, we have:
$$x + 3x - 36 = 180$$
Simplifying, we get:
$$4x = 216$$
$$x = 54$$
Thus, if angle $\alpha$ and angle $\beta$ are supplementary, $\angle \beta = 54^\circ$. Substituting $x = 54$ into the expression for $\angle \alpha$, we find:
$$\angle \alpha = 3 \times 54^\circ - 36^\circ = 162^\circ - 36^\circ = 126^\circ$$

Therefore, the degree of angle $\alpha$ can be either $18^\circ$ or $126^\circ$. Hence, the final answer is encapsulated as $\boxed{18^\circ \text{ or } 126^\circ}$.