sorry my original answer was wrong.
how's this sound:
I'll call the vectors x,y,z
first find the normalized vector for each vector.
we have 3 unique sides, the xy, xz, yz sides. We'll find the area for those then multiply by 2.
Lets' work xy and the others will be obvious.
Pick one vector to be the base lets choose x. The length of this vector with be the width of the parallelogram. We need the height.
now we calculate the angle between x and y
(that's a dot product, not a multiply)
now the height of the parallelogram is
you can do the same for xz and yz sides
how's that look?