The parallelogram could be a rectangle with side ratios 9:11, in which case the angles of the parallelogram are all 90 degrees. So that means there's not enough information in the question to find a unique solution.
Let's say the actual lengths of the sides are a and b, and the lengths of the diagonals are c and d. If x is the angle of the parallelogram, we use the cosine rule:
c^2=a^2+b^2-2abcosx and d^2=a^2+b^2+2abcosx because the adjacent interior angles of a parallelogram are complementary. If we subtract these equations we get: d^2-c^2=4abcosx. So cosx=(d^2-c^2)/4ab. To find x we need to know the lengths of a, b, c, d or some ratios. One ratio isn't enough. So we could use the ratio of the lengths of the diagonals, the ratio of the sides, and the ratio of a side to a diagonal. We can rewrite this equation: d^2(1-c^2/d^2)/(4b^2(a/b))=(d^2/b^2)(1-c^2/d^2)/(4(a/b))=cosx; or (d^2/b^2-c^2/b^2)/(4(a/b))=cosx. If we use r1, r2 and r3 for the ratios, we get (r1^2-r2^2)/4r3=cosx. Here we see ratios r1=d/b, r2=c/b or r2/r1=c/d, and r3=a/b, but we only know one: r3=a/b=9/11. We can also see that if c=d, cosx=0 and x=90 degrees (rectangle has diagonals of equal length).