Calculating area of a parallelogram using length of two diagonals Is there a formula for calculating area of a parallelogram using only the length of diagonals? if so what is it?
No.  Imagine two fixed diagonals that are attached and can rotate in the center.  As they pivot, different parallelograms are induced; if they are almost parallel the area will be close to zero.
If the diagonals are given in vector form, then as @RecklessReckoner said, one diagonal is vector sum $a+b$ and the other is vector sum $a-b$. So you have two equations and two variables, you can find out vectors $a$ and $b$, then simply find the magnitude of their cross product.

If the diagonals are vectors $p$ and $q$, the area can be represented as the magnitude of the cross product: $\frac 14 (p+q) \times (p-q)$.