Let ABCD be the parallelogram circumscribing the circle such that:

AB is tangent to the circle at P,

BC is tangent to the circle at Q,

CD is tangent to the circle at R and

DA is tangent to the circle at S.

AP = AS (they are tangent segments to the same circle from the same exterior point hence they are equal)

Similarly,

BP = BQ

CR = CQ and

DR = DS

Adding all of the above 4 equations we get:

AP + BP + CR + DR = AS + BQ + CQ + DS

AP + PB + CR + RD = BQ + QC + AS + SD

AB + CD = BC + AD

But AB = CD and BC = AD (Opposite sides of a parallelogram are equal)

If we plug AB = CD and BC = AD into AB + CD = BC + AD we get:

2AB = 2BC divide both sides by 2

AB = BC

Hence AB = BC = DC = DA

Thus all the 4 sides are equal hence the parallelogram ABCD is a rhombus.