Two lines

$\overrightarrow{r}={\overrightarrow{a}}_1+\lambda {\overrightarrow{b}}_1$ and $\overrightarrow{r}={\overrightarrow{a}}_2+\lambda {\overrightarrow{b}}_2$

are parallel if ${\overrightarrow{b}}_1=m{\overrightarrow{b}}_2, \mathrm{for} m\in R$

and perpendicular if ${\overrightarrow{b}}_1·{\overrightarrow{b}}_2=0$

and intersects, if distance between them $=0$.

We have lines $L_1 :=(\hat{i}+5\hat{j}+5\hat{k})+t(4\hat{i}-4\hat{j}+5\hat{k})$ and 

$L_2 : \overrightarrow{r}=(2\hat{i}+4\hat{j}+5\hat{k})+s(8\hat{i}-3\hat{j}+\hat{k})$,

Here we can see, $4\hat{i}-4\hat{j}+5\hat{k}\ne$$m(8\hat{i}-3\hat{j}+\hat{k})$ for $m\in R$.

Hence, these two lines are not parallel.

Checking perpendicularity: 

 $\left(4\hat{i}-4\hat{j}+5\hat{k}\right)·(8\hat{i}-3\hat{j}+\hat{k})=32+12+5\ne 0$

So, they are non-perpendicular.

Hence, these lines are either Skew lines or intersect each other.

Finding the distance between them, 

$d=\left|\frac{\left[(2\hat{i}+4\hat{j}+5\hat{k})-(\hat{i}+5\hat{j}+5\hat{k})\right]·\left[(8\hat{i}-3\hat{j}+\hat{k})\times (4\hat{i}-4\hat{j}+5\hat{k})\right]}{(8\hat{i}-3\hat{j}+\hat{k})\times (4\hat{i}-4\hat{j}+5\hat{k})}\right|\ne 0$

Hence, both are Skew lines, distance between them $\ne 0$