Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$.  The three lines are distinct and lie in a plane.  The number of points equidistant from all three lines is:

$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
1. Let's denote the lines as follows:
   - \( l_1 \) and \( l_3 \) are parallel lines.
   - \( l_2 \) intersects both \( l_1 \) and \( l_3 \).

2. Since \( l_3 \) is parallel to \( l_1 \), the distance between \( l_1 \) and \( l_3 \) is constant. Let this distance be \( d \).

3. Consider the line \( m \) that is equidistant from both \( l_1 \) and \( l_3 \). This line \( m \) is parallel to both \( l_1 \) and \( l_3 \) and is exactly halfway between them. Therefore, the distance from \( m \) to \( l_1 \) and \( l_3 \) is \( \frac{d}{2} \).

4. Now, we need to find the points that are equidistant from \( l_1 \), \( l_2 \), and \( l_3 \). These points must lie on the line \( m \) and also be equidistant from \( l_2 \).

5. Since \( l_2 \) intersects \( l_1 \) and \( l_3 \), it will also intersect the line \( m \) at some point. Let this intersection point be \( P \).

6. The points equidistant from \( l_1 \), \( l_2 \), and \( l_3 \) must lie on the perpendicular bisector of the segment formed by the intersection points of \( l_2 \) with \( l_1 \) and \( l_3 \). This perpendicular bisector will intersect \( m \) at two points, one on each side of \( P \).

7. Therefore, there are exactly two points on the line \( m \) that are equidistant from \( l_1 \), \( l_2 \), and \( l_3 \).

Conclusion:
The number of points equidistant from all three lines is \( \boxed{2} \).