Are all squares parallelograms?

Yes. But not all parallelograms are squares.

All squares are also rectangles (and all rectangles are parallelograms), but not all rectangles are squares.

"parallelogram" just means both pairs of sides are parallel (and rectangle just means all the angles are right). Yes, since they all have a pair of sides that are parallel, they are all parallelograms. Parallelograms are squares only if all four sides are of the same length and all interior angles are 90o.