jmc

We count the number of rectangles by cases, based on the side lengths of the rectangle: $$\begin{array}{cc}\text{Side lengths of rectangle}& \text{Number of rectangles}\\ 1\times 1& 9\\ 1\times 2& 6\\ 1\times 3& 3\\ 2\times 1& 6\\ 2\times 2& 4\\ 2\times 3& 2\\ 3\times 1& 3\\ 3\times 2& 2\\ 3\times 3& 1\end{array}$$ So the number of rectangles whose sides are parallel to the sides of the grid is $9+6+3+6+4+2+3+2+1=\boxed{36}.$

Extra challenge: If you know what "combinations" are in counting problems, try to find a much faster solution!