I think you're on the right track with your parametric equation.
What you have there is the vector form of the line equation.
L = R + tV
Where R is [x0, y0, z0] and V is [a, b, c].
You just need to normalize your equations. You would do that by finding the value of R such that |R| is at a minimum, which occurs when R is perpendicular to L, or R.L = 0.
Also, since t can be scaled by any value, without changing the line, you should normalize V by dividing every coefficient by |V|