Can a trapezoid and a parallelogram with the same base and height have the same area

No, unless you call a rectangle a trapezoid.

Recall the formula for the area of a parallelogram: .  Recall the area for a trapezoid: (a trapezoid has two bases).

Suppose they did have the same area.  Then and and .

So the only way for their areas to be equal is for the trapezoid to have the same base on top and bottom.  That means the trapezoid must be a rectangle.  Most people don't say a rectangle is a trapezoid, so no.