Question

Can the quadrilateral be a parallelogram if two opposite angles are right angles, but the quadrilateral is not a rectangle?

Answer 1

Geometry.
If two opposite angles are right angles, then the other angles have to be right angles. So yes, it is a parallelogram.

Answer 2

If the two opposite angles are 90 degrees(right angles) the other two angles must also be right angles.
If it's not a rectangle it has to be a square.