Main Difference
The main difference between Parallelogram and Quadrilateral is that the Parallelogram is a quadrilateral with two pairs of parallel sides and Quadrilateral is a polygon with four sides

Parallelogram
In Euclidean geometry, a parallelogram is a simple (nonselfintersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.
By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.
The threedimensional counterpart of a parallelogram is a parallelepiped.
The etymology (in Greek παραλληλόγραμμον, a shape “of parallel lines”) reflects the definition.

Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5sided), hexagon (6sided) and so on.
The origin of the word “quadrilateral” is the two Latin words quadri, a variant of four, and latus, meaning “side”.
Quadrilaterals are simple (not selfintersecting) or complex (selfintersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is
∠
A
+
∠
B
+
∠
C
+
∠
D
=
360
∘
.
{displaystyle angle A+angle B+angle C+angle D=360^{circ }.}
This is a special case of the ngon interior angle sum formula (n − 2) × 180°.
All nonselfcrossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Parallelogram (noun)
A convex quadrilateral in which each pair of opposite edges are parallel and of equal length.

Parallelogram (noun)
either of two rectangular areas (respectively the large parallelogram and the small parallelogram) abutting the goal line in front of the goal. (Since 1986 officially named the large rectangle and small rectangle, though the older names are still occasionally used.)

Quadrilateral (noun)
A polygon with four sides.

Quadrilateral (noun)
An area defended by four fortresses supporting each other.
“The Venetian quadrilateral comprised Mantua, Peschiera, Verona, and Legnano.”

Quadrilateral (adjective)
having four sides.