Parallel lines would always lie in the same plane. They would need to be skew lines.
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They're either parallel lines or skew lines.
-- An infinite number of different planes can intersect the same line. -- The same line can lie in an infinite number of different planes. -- An infinite number of different lines can intersect the same plane.
Skew lines are lines that lie on separate planes. and do not intercept parallel lines that are sometimes on the same plane, but don't intersect
skew lines
Parallel Lines - Skew lines are lines that do not intersect and do not lie in the same plane. If two lines lie in the same plane they they must either intersect or they are parallel.