Some do, but not all. It's not a general thing
that you can say about all parallelograms.
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A parallelogram normally has no lines of symmetry unless it is in the shape of a rectangle which will then give it 2 lines of symmetry
A trapezoid is a 4 sided quadrilateral with no lines of symmetry unless it's an isosceles trapezoid Also parallelograms have no lines of symmetry unless it is a rectangle
Circles and Ovals DO have lines of symmetry: a circle has an infinite number of them (each is a diameter of the circle) and an oval (ellipse) has two (one along the major axis, one along the minor axis). Shapes which have no lines of symmetry are irregular ones, eg scalene triangles, along with most parallelograms (ie parallelograms which are not rhombuses) and non-isosceles trapezia. Some irregular shapes can have lines of symmetry, eg irregular octagons can have 1, 2 or 4 lines of symmetry as well as no lines of symmetry, unlike a regular octagon which [always] has 8 lines of symmetry.
Yes rectangles and parallelograms have rotation symmetry. rectangles and parallelograms have a rotational symmetry of 2. I hope this information helps you.
No ! a parallelogram doesn't have any line of symmetry !