6. There exist ( ) pairs of parallel lines such that no line can intersect both at the same time.
A. 2
B. 3
C. 4
D. None of the above
6. C

Obviously, exclude A.
For B, for any 3 pairwise skew lines, there are infinitely many lines intersecting them (refer to the 1997 National League).
For C, the proof is as follows:
Take the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, then the lines $A B, A_{1} C, D_{1} D, B_{1} C_{1}$ are pairwise skew lines. Assume there exists a line $l$ intersecting line $A B$ at $E$, line $A_{1} C$ at $F$, line $D_{1} D$ at $G$, and line $B_{1} C_{1}$ at $H$.

Now rotate the entire figure around $A_{1} C$ by $120^{\circ}$ (as shown in the figure).

Since $A_{1} C$ is perpendicular to $A B_{1} D_{1}$ and passes through the center of the equilateral $\triangle A B_{1} D_{1}$, point $A$ becomes point $B_{1}$, point $B_{1}$ becomes point $D_{1}$, point $D_{1}$ becomes point $A$; similarly, point $B$ becomes point $C_{1}$, point $C_{1}$ becomes point $D$, point $D$ becomes point $B$, i.e., line $A B$ becomes line $B_{1} C_{1}$, line $B_{1} C_{1}$ becomes line $D_{1} D$, line $D_{1} D$ becomes line $A B$, the positions of the four lines in space remain unchanged, only the labels are cyclically permuted.

At this time, the line $L$ passing through $F$ and rotating around $A_{1} C$ by $120^{\circ}$ cannot coincide with itself $L$, denoted as $L_{1}$; thus, the new position $G_{1}$ of point $G$ on $A B$ cannot coincide with $E$, otherwise, by the fact that two points determine a line, $L$ (which is $F E$) and $L_{1}$ (which is $F G_{1}$) would coincide, leading to a contradiction. Similarly, the new position $H_{1}$ of point $H$ on $D_{1} D$ does not coincide with $G$, thus, the coplanar lines $L, L_{1}$ intersecting at $F$ intersect $A B$ at $E, G_{1}$, and $D_{1} D$ at $G, H_{1}$, making $E, G_{1}, G, H_{1}$ coplanar, i.e., lines $A B$ and $D D_{1}$ are coplanar, which contradicts the fact that $A B, D D_{1}$ are skew lines.
Therefore, a line $L$ intersecting $A B, A_{1} C, D_{1} D, B_{1} C_{1}$ does not exist.