Parallel structure is not part of grammar, except to the extent that there are syntactic rules, like Conjunction Reduction, that apply to structures that are syntactically parallel. They're always optional rules, though, and there's no grammatical requirement to use parallel structures. They're awfully useful, though, as you point out
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Two lines will remain parallel when they are intersected by a transversal line
the reason their are parallel lines because the person who invented them thought they were compulsory for maths
Train tracks
It is not because it does not have two pairs of parallel sides.
It is possible, but not necessary. A hexagon can have 0, 2, 3, 4, 5, or 6 parallel sides. A regular hexagon has 3 parallel sides.