Parallel lines are lines that are perfectly aligned with each other. This alignment is accomplished by having each line change at the same rate; or in other words having the same slope. Liner functions are most popularly known so I will describe linear alignment. Most people know the function y=mx+b where y and x are the axis, m is the slope, and b is the y-intercept.
Slope (m) is also known as rate of change which is what allows lines to be parallel. Slope allows this because going from any points if they change at the same rate then the next set of points will be equidistant as the first set of numbers; and we can apply this principle all the way from negative infinity to positive infinity over a set of all real numbers.
For example lets have two lines y=3x+1 and y=3x-99. If we plug in zero for both lines we will get points at (0, [3(0)+1]) or (0,1) for Line 1; and (0,[3(0)-99]) or (0,-99) for Line 2. Then we can check the distance between these points which is 1-(-99) = 100. Then if we move over 1 x unit and plug in that we get points (1, [3(1)+1]) or (0,4) for Line 1; and (1,[3(1)-99]) or (0,-96) for Line 2. Now if we check the distance between these new points we will get 4-(-96) = 100. Now that is interesting that these points are the same distance no matter the x-value and that is because the m value is similar; which means if any function where the leading coefficients of each x-value are the same as another we can say they are parallel.
Now that we know how these parallel lines work we can look further into the applications and uses of this alignment. We can use alignment in geometry and in particular cases in calculus. In geometry we if we know two lines are parallel we can drag any segment through these two lines and know all of the angles; this is useful when we are finding relationships between triangles and parallelograms. We also can flatten functions out so they are horizontal and much easier to calculate.
In calculus and physics we are always trying to linearize any function we are given to make them simpler to understand. To do this we transform our functions; and take derivatives to make tangent lines. In more advanced mathematics we have many uses for parallel lines like finding congruencies in statistical plots but in general we can find that a parallel relationship between functions and lines are a very important concept in every topic of mathematics.