Which area is more fundamental: parallelogram or triangle?

The intuitive way, as you were probably taught as a child, is to count squares on graph paper. The next step is to draw the formula for the rectangle, $a=w\times h$
Neither triangle nor parallelogram comes first, and this is because they require three directions (or three lengths), when the rectangle only needs two.
Also note that the area of the triangle under the form ${\displaystyle \frac{w\times h}{2}}$ is "teachable", while Heron's formula $\sqrt{s(s-a)(s-b)(s-c)}$ cannot be explained early. This said, an important formula is the shoelace, that computes the area of an arbitrary polygon. It is based on the area of… the trapezoid.