That's WRONG!  Look at the black parallelogram below.  The red figure is formed
by cooecting the midpoints of the sides of the black parallelogram.  But as you
can see, the red figure is not a rhombus (diamond-shape).  If it were a rhombus,
then its two diagonals (in green) would be perpendicular, and it's obvious that
they are not. 

All you can say is that it is another parallelogram, and not any special
kind of parallelogram, necessarily.  



Edwin

Continued from above. (Edwin McCravy and AnlytcPhil are the same person!) 
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Let's look at a few related problems to the above while we're at it:

If the black parallelogram were a rhombus, then the red
one would be a rectangle, like this shows:



And if the black one were a rectangle, the red one would be a rhombus,
like this shows:



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And here's the one I find most interesting.  Even if the black figure is
not any special quadrilateral at all, like the black quadrilateral
below, then the red figure is always a parallelogram, like this shows!: