A rhombus is a quadrilateral whose four sides all have the same length.
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A square is a rhombus by the modern definition of rhombus as a quadrilateral with four equal sides. An old definition of rhombus added the clause that it is not a square.
A rhombus is always a parallelogram, by definition.
A rhombus by definition has four sides, and a triangle has only three. So, a triangle is never a rhombus.
A rhombus, by definition, must be equilateral. Therefore, an isosceles rhombus cannot exist and so it cannot be a parallelogram.
Look at the definition of a rhombus. A square fulfills all parts of the definition.