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Written solution:

The two chords of length $38$ are equally far from the center of the circle. Because the three parallel lines are equally spaced, those two equal chords must lie on adjacent lines, with the center halfway between them. Let that half-distance be $d$. Then each $38$-chord is distance $d$ from the center, and the $34$-chord is distance $3d$ from the center.

If the circle has radius $r$, then

$$r^2=19^2+d^2=17^2+(3d)^2.$$

Thus $19^2-17^2=8d^2$, so $72=8d^2$, and $d=3$. The distance between adjacent parallel lines is $2d=6$.

Thus, the answer is B .