Is flow between parallel plates rotational?

A flow is irrotational if the rotation (half the vorticity) is zero everywhere: $\omega_z = \dfrac{1}{2}\left(\dfrac{\partial v}{\partial x} - \dfrac{\partial u}{\partial y}\right) = 0$.

Given: $u = \dfrac{1}{2\mu}\dfrac{dP}{dx}(y^2 - hy)$, $v = 0$.

$$\frac{\partial v}{\partial x} = 0,\qquad \frac{\partial u}{\partial y} = \frac{1}{2\mu}\frac{dP}{dx}(2y - h)$$ $$\omega_z = \frac{1}{2}\left(0 - \frac{1}{2\mu}\frac{dP}{dx}(2y - h)\right) = -\frac{1}{4\mu}\frac{dP}{dx}(2y - h)$$

The vorticity is $\zeta = 2\omega_z = -\dfrac{1}{2\mu}\dfrac{dP}{dx}(2y - h)$.

Since $dP/dx$ is a non-zero constant, $\omega_z$ is zero only at the mid-plane $y = h/2$ and non-zero elsewhere. Because $dP/dx < 0$, $\omega_z$ has the sign of $(2y - h)$: it is negative near the bottom plate ($y < h/2$) and positive near the top plate ($y > h/2$).

Answer: the flow is rotational, because $\omega_z \neq 0$ everywhere except along the centreline $y = h/2$. The rotation arises from viscous shear (the velocity gradient $\partial u/\partial y$), which is largest at the walls.