Parallelograms: 2-fold
Square: 4-fold
n-fold symmetries refer to rotational symmetries. Consequently, any symmetries about axes that these and other quadrilaterals may have are not relevant to this question.
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Not all 4 sided quadrilaterals have lines of symmetry.
Not all 4 sided quadrilaterals have lines of symmetry although some of them do have lines of symmetry.
No.
A square
Rhombus