How can I tell if this parametric equation intersects?
Each of them $(X_1=X_2, Y_1=Y_2, Z_1=Z_2)$ are equal to $0$. Does this mean they do not intersect? Are they parallel?
I am willing to clear things up need it be, of if there's any information currently missing that you'd like to know. Thanks in advance.
The answer is yes, they will intersect. Actually, your starting point is correct, but you need to define your problem in a mathematical way.
In brief, if there exists a pair $(t,u)$ that makes $x(t)=x(u)$ and the same for $y,z$, then you can claim they intersect. Otherwise, they don't. To be more specific, if there is a solution for
\begin{align} t+5&=4u+11\\ 2t-6&=-2u-4\\ 3t-1&=6u+11, \end{align} then these two lines intersect. The solution can be easily obtained as $(t,u)=(2,-1)$. So there is an intersection.
To check if two lines are parallel, you can wrap it up as another math problem that if $\frac{d}{dt}x(t)=|\frac{d}{du}x(u)|$ and the same for $y,z$. If the answer is yes, then these two lines are parallel.