Is this true or false: A system comprising two linear equations in two variables has a unique solution if and only if the straight lines represented by the equations are nonparallel?

true.

2 Line equations will have a unique solution if and only if they
intersect each other at a particular point ( x, y ). If Lines are
parallel, then they'll never intersect and hence we can not find any solution.

if we consider two parallel Lines of equations,say-

2x + 3y = 7 ,  2x + 3y = 4                ( slope of both equation are same )

now, clearly these equation have no solution. So it is true that A
system comprising two linear equations in two variables has a unique
solution if and only if the straight lines represented by the equations
are nonparallel