if two lines are parallel, do they determine a plane? (is there one, and only one plane which contains the lines)

Four ways to determine a plane:

1) two parallel lines
2) two intersecting lines
3) three noncollinear points
4) a line and a point not on the line

No, two parallel lines do not determine a plane. In Euclidean geometry, a plane is determined by three non-collinear points. Collinear points are points that lie on the same line, so if you have two parallel lines, they will never intersect and thus will not provide a third non-collinear point to determine a plane.

To see this conceptually, imagine having two straws on a table, perfectly parallel to each other. You can try placing a flat sheet of paper over the straws to form a plane. But regardless of how you arrange the paper, it will always touch the straws at two distinct points and will never be able to lie flat between them.

So, to determine a plane in this case, you would need at least one more line that is not parallel to the given lines. Only then will you have three non-collinear points, which can be used to define a unique plane containing all three lines.