Except there is something called an "isosceles trapezoid" (or isosceles trapezium--depending on where you live), which is not a parallelogram, because it has line symmetry.
Either way you think of a trapezoid, it's a question of arbitrary definition, not mathematical principle, which is why we think the inclusion of something so low level as a testable element of Common Core (a trapezoid is first mentioned in 1st grade--will they test that?) is just another of an endless series of dumbed down standards.
Using one definition over the other is also not being honest to students, a practice we abhor.
Sorry to those educators who have to spend time worrying about such issues, which could be time spent on bona fide mathematical questions instead.