Let's find the gradients of the two lines:
y=(13-3x)/7 and y=(zx+8)/3. The gradients are -3/7 and z/3.
To be parallel, these gradients must be the same so z/3=-3/7, z=-9/7.
[This would make the second equation -9x/7+8=3y or -9x+56=21y which can also be written 9x+21y=56.]