1. Let the line intersect the parabola $y = x^2$ at two lattice points $(x_1, x_1^2)$ and $(x_2, x_2^2)$. The line also passes through the point $(0, 2016)$.

2. The equation of the line passing through $(0, 2016)$ and $(x_1, x_1^2)$ can be written as:
$$y - 2016 = m(x - 0) \implies y = mx + 2016$$
where $m$ is the slope of the line.

3. Since the line also passes through $(x_2, x_2^2)$, we have:
$$x_2^2 = mx_2 + 2016$$

4. The slope $m$ can be determined by the two points $(x_1, x_1^2)$ and $(x_2, x_2^2)$:
$$m = \frac{x_2^2 - x_1^2}{x_2 - x_1} = x_1 + x_2$$
Therefore, the equation of the line becomes:
$$y = (x_1 + x_2)x + 2016$$

5. Substituting $y = x_1^2$ into the line equation when $x = x_1$:
$$x_1^2 = (x_1 + x_2)x_1 + 2016$$
Simplifying, we get:
$$x_1^2 = x_1^2 + x_1 x_2 + 2016 \implies x_1 x_2 = -2016$$

6. We need to find pairs $(x_1, x_2)$ such that $x_1 x_2 = -2016$ and both $x_1$ and $x_2$ are integers.

7. The number 2016 can be factored into its prime factors:
$$2016 = 2^5 \cdot 3^2 \cdot 7$$
The total number of divisors of 2016 is:
$$(5+1)(2+1)(1+1) = 6 \cdot 3 \cdot 2 = 36$$

8. Since $x_1 x_2 = -2016$, for each positive divisor $d$ of 2016, there is a corresponding negative divisor $-\frac{2016}{d}$. Thus, there are 36 pairs $(x_1, x_2)$ such that $x_1 x_2 = -2016$.

9. Each pair $(x_1, x_2)$ corresponds to a distinct line passing through $(0, 2016)$ and intersecting the parabola $y = x^2$ at two lattice points.

Therefore, the number of distinct lines is $\boxed{36}$.