jmc

Since $V$ is the midpoint of $PR,$ then $PV=VR.$ Since $UVRW$ is a parallelogram, then $VR=UW.$ Since $W$ is the midpoint of $US,$ then $UW=WS.$

Thus, $$PV=VR=UW=WS.$$ Similarly, $$QW=WR=UV=VT.$$ Also, $R$ is the midpoint of $TS$ and therefore, $TR=RS.$ Thus, $△VTR$ is congruent to $△WRS$, and so the two triangles have equal area.

Diagonal $VW$ in parallelogram $UVRW$ divides the area of the parallelogram in half. Therefore, $△UVW$ and $△RWV$ have equal areas.

In quadrilateral $VRSW,$ $VR=WS$ and $VR$ is parallel to $WS.$ Thus, $VRSW$ is a parallelogram and the area of $△RWV$ is equal to the area of $△WRS.$ Therefore, $△VTR,$ $△WRS,$ $△RWV,$ and $△UVW$ have equal areas, and so these four triangles divide $△STU$ into quarters.

Parallelogram $UVRW$ is made from two of these four quarters of $△STU,$ or one half of $△STU.$ The area of parallelogram $UVRW$ is thus half of $1,$ or $\boxed{\frac{1}{2}}.$