"Parallelograms only exist in Euclidean Geometry. False?"
No, true because the sum of the angles in other geometries are either more or less than 360 degrees.
If a theorem is proved using the Euclidean Parallel Postulate or its equivalence, then it is strictly Euclidean. True?
Parallelograms only exist in Euclidean Geometry. False?
A quadrilateral that is both Saccheri and Lambert is a rectangle. False?
Maybe I'm being too restrictive. Anyways I found this for you through a Google search:
http://public.csusm.edu/aitken_html/m410/quad.pdf
Then what do you mean by "parallelogram"? Or, for that matter, by "a parallel"? If you take any two non-intersecting lines to be "parallel", then, yes, a Saccheri Quadrilateral is a parallelogram. But many texts make a distinction between "parallel lines" (limiting parallels in other books) and "non-intersecting lines". Opposite sides of a Saccheri quadrilateral are non-intersecting lines.