I'm not sure I understand the question! "Explain what this tells you about parametric and symmetric equations in R^3." It doesn't tell you anything about them in general! Did you mean, "Explain what this tells you about parametric and symmetric equations of this line in R^3"?
If a line is parallel to the xy-plane, then z does not change. The parametric equations will have z= constant. The requirement that it not be parallel to any axis just says that x and y do change. Typically the way you form the "symmetric equations" from the parametric equations is to solve each equation for the parameter, then set them all equal. Since here, z is a constant, there is no parameter in that equation and so you cannot solve for it. The "symmetric equations" will be just a single equation in x and y, with the additional equation z= constant. Geometrically, you can think of this as the a line in xy-plane "shifted" up or down along the z-axis.
It's easy to make up examples. Just write "x= f(t), y= g(t), z= c" where f(t) and g(t) are any linear functions you want to use and c is any constant.