The y-intercept is where a line crosses the y-axis, which is when x = 0.
Parallel lines on a graph mean that two or more lines have the exact same slope as one another (which is what makes them parallel: they cannot touch, and if their slopes differ, they eventually will sooner or later). The y-intercept is a tool to see just how close or how far each line is to one another:
y = 3x + 5
y = 3x + 4
y = 3x + 6,783,648,213,464,987,123
Each of these lines are parallel: they each have the exact same slope, so no matter what value up put into x, they will never touch one another.
When we put 0 into the value of x, we see where they cross the y-axis--find the y-intercept:
y = 3 * 0 + 5
y = 0 + 5
y = 3 * 0 + 4
y = 0 + 4
y = 3 * 0 + 6,783,648,213,464,987,123
y = 0 + 6,783,648,213,464,987,123
The y-intercept, thus, is a tool to help define a point on the line to determine where on the graph it resides.