$A\left(1,2,3\right),B\left(4,5,7\right),C\left(-4,3,-6\right),D\left(2,k^2,2\right)$ are four points. If the lines $\overrightarrow{AB}\text{ and }\overrightarrow{CD}$ are parallel, then $k=$

If two lines are parallel, then their direction ratios are in proportional 

The direction ration of line joining $A\left(x_1,y_1,z_1\right)\text{ and }B\left(x_2,y_2,z_2\right)$ is $〈x_2-x_1,y_2-y_1,z_2-z_1〉$

The direction ration of line joining $A\left(1,2,3\right)\text{ and }B\left(4,5,7\right)$ is$〈3,3,4〉$

The direction ration of line joining $C\left(-4,3,-6\right)\text{ and }D\left(2,k^2,2\right)$ is$〈6,k^2-3,8〉$

Since the lines are parallel, their direction ratios are proportional. It implies that$\frac{6}{3}=\frac{k^2-3}{3}=\frac{8}{4}$

Therefore, $k^2-3=6\Rightarrow k^2=9\Rightarrow k=\boxed{\pm 3}$