How do you find the equation of parallel lines?

The equation of a straight line is y = mx + c

Parallel lines will have the same gradient, so the value of m is the same for each line

If you're just finding "any" line parallel to an original, pick any value of c. If you need parallel lines passing through a particular point then plug the coordinates of that point into the straight line equation with the value of m that you already found and rearrange to find c.