Jacobian of affine warp in Lucas-Kanade image alignment
You're right that all the quantities are computed for a single pixel, so in the product

$$\nabla I\cdot \frac{\partial W}{\partial p}\cdot\Delta p\tag{1}$$

the sizes of the vectors and matrices are

$$\nabla I\quad\;\;\; 1\times 2\\
\frac{\partial W}{\partial p}\quad 2\times n\\
\Delta p\quad\;\;\;\; n\times 1$$

where $n$ is the number of parameters. So the expression in (1) is a scalar. The total error measure given by Equation (6) in the paper is the sum of this scalar expression over all pixels.