No, a parallelogram has two dimensions.
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No. A cube is a 3-dimensional object while a parallelogram is 2 dimensional.
No. A triangle, like a circle, square, parallelogram, or trapezoid are all constructs upon a single 2-dimensional plane.
A parallelogram is a two-dimensional figure, so it has two dimensions.
Yes. Parallelograms are flat shapes (polygons). 3-dimensional forms are polyhedrons.
A parallelopiped has six faces that are parallelogram - like a cuboid that has been distorted by pushing opposite vertices in opposite directions (the three dimensional equivalent of distorting a rectangle into a parallelogram). A solid with two rectangular faces would be a special case of a parallelopiped.