This does not mean that the parcel is not adiabatic, though. For a parcel to be truly adiabatic, $\frac{\partial \theta}{\partial t}+u\frac{\partial \theta}{\partial x}+v\frac{\partial \theta}{\partial y}+w\frac{\partial \theta}{\partial z}=0$ Therefore, the only time a certain location can be considered adiabatic and have $\frac{\partial \theta}{\partial z}=0$ is when there is no heating ($\frac{\partial \theta}{\partial t}=0$), and no horizontal advection of $\theta$ ($u\frac{\partial \theta}{\partial x}+v\frac{\partial \theta}{\partial y}=0$). Note, I did not exclude vertical advection of $\theta$, because that is implied in the condition that $\frac{\partial \theta}{\partial z}=0$.

I'll also take pause here and list a few diabatic processes, and why there may be some justification to ignore them for your mental image.

• Radiation: with the exception of greenhouse gases, the atmosphere is mostly transparent.
• Surface heating/ turbulent transport: more important near the surface (note that $\theta$ is a minimum at the surface).
• Clouds/microphysical processes: discrete and are never everywhere all the time and moisture is a completely separate variable that varies all the time. There is a way around that (to some extent- the equivalent potential temperature, for latent heat exchange).
• Chemical processes: See below about how I remember $\theta$ increases with height.

If you want a good reminder of why $\theta$ increases with height, it might be easier to remember that $sgn \left( \frac{\partial \theta}{\partial z}\right)$ is a good indicator of static stability. And thunderstorms need to stop at some point, even if that is the tropopause (where $\frac{\partial \theta}{\partial z}>0$ due to ozone heating).