A rectangle is a quadrilateral with four right angles, so while it is useful to employ the idea of a parallelogram, it isn't necessary.
The reason is in the unstated part of both your proposals: what connects a single right angle to all angles being right?
If you show a parallelogram, you can use the properties of a parallelogram: opposite angles are equal, adjacent angles are supplementary.
If you don't show a parallelogram, you can use the property of transversals across parallel lines: corresponding angles are supplementary (which you have to iterate one extra step for the opposite angle, or rely on a quadrilateral - 3 right angles = 1 right angle).
So in my assessment, it is easier to apply the higher-order properties of parallelograms than the more fundamental transversal properties, but the difference is in this step of the proof.
An even simpler approach is this:
Hope this helps,
PS: Check back at this URL in a couple of days; if other math consultants send other ideas, I'll post them here.