Main Difference
The main difference between Parallelogram and Rhombus is that the Parallelogram is a quadrilateral with two pairs of parallel sides and Rhombus is a quadrilateral in which all sides have the same length

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.

Rhombus
In Euclidean geometry, a rhombus (◊) (plural rhombi or rhombuses) is a simple (nonselfintersecting) quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a diamond, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is a parallelogram and a kite. A rhombus with right angles is a square.

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.)

Rhombus (noun)
Any of several flatfishes, including the brill and turbot, once considered part of the genus Rhombus, now in ver=170601. from 16th c.

Rhombus (noun)
Snails, now in genus Conus or family Conidae.

Rhombus (noun)
A parallelogram having all sides of equal length. from 16th c.

Rhombus (noun)
a quadrilateral all of whose sides have the same length.