Question

can a trapezoid and a parallelogram with the same base and height have the same area ?

Answer 1

No, a trapezoid has both sides angling inward, while the parallelogram only has one going inward (toward the top).

Answer 2

the two bases of a parallelogram are equal
the two bases of a trapezoid are different

Bh vs (b+B)h/2

These can only be the same if b=B.