Ok we have the equation:
[tex]\begin{gathered} y-\frac{1}{3}x=-2 \\ y=\frac{1}{3}x-2 \end{gathered}[/tex]What we have here is a linear function that is defined by two parameters: the slope that is the number multiplying x (1/3 in this case) and the y-intercept that is the independent term (here is -2). Two lines are parallel if they have the same slope independently of the value of the y-intercept.